'Pitt 


O.A.C. 


ELECTRIC 
POWER  TRANSMISSION 


PUBLISHERS     OF    BOOKS     F  O  R^. 

Electrical  World  v  Engineering  News-Record 
Power  v  Engineering  and  "Mining  Journal-Press 
Chemical  and  Metallurgical  Engineering 
Electric  Railway  Journal  v  Coal  Age 
American  "Machinist v  Ingenieria  Internacional 
Electrical  Merchandising  ^  BusTransportation 
Journal  of  Electricity  and  Western  Industry 
Industrial  Engineer 


ELECTRIC   POWER 
TRANSMISSION 

PRINCIPLES  AND  CALCULATIONS 

INCLUDING  A  REVISION  OF 
"OVERHEAD  ELECTRIC  POWER  TRANSMISSION" 


BY 
ALFRED  STILL 


PROFESSOR  OF   ELECTRICAL  DESIGN,  PURDUE  UNIVERSITY;    MEMBER  OF  THE  INSTITUTION   OF 

ELECTRICAL  ENGINEERS;  FELLOW  OF  THE  AMERICAN  INSTITUTE  OF  ELECTRICAL 

ENGINEERS;    MEMBER    OF   THE   INSTITUTION   OF   CIVIL   ENGINEERS; 

AUTHOR  OF  "  POLYPHASE  CURRENTS;"  "  PRINCIPLES 

OF  ELECTRICAL  DESIGN;"  ETC. 


SECOND  EDITION 

REVISED,  ENLARGED  AND  REWRITTEN 
FIFTH  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON:  6  &  8  BOUVERIE  ST.,  E.  C.  4 
1919 


COPYRIGHT,  1913,  1919,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


PRINTED  IN  THE    UNITED   STATES    OF  AMERICA 


TK 


PREFACE  TO  SECOND  EDITION 

When  this  book  was  published  originally  under  the  title 
"Overhead  Electric  Power  Transmission,"  its  suitability  for  use 
as  a  College  text  had  not  been  seriously  considered.  It  has, 
however,  been  used  by  a  large  number  of  technical  schools  and 
colleges,  and  while  the  restricted  scope  of  the  book  may  limit 
its  suitability  as  a  text  for  college  students,  the  changes  and 
additions  which  will  be  found  in  this  new  edition  should,  in  the 
author's  opinion,  enhance  its  value  as  a  college  text  without 
detracting  from  its  usefulness  in  the  field  of  practical  engineering. 

The  principal  addition  —  which  has  necessitated  an  alteration 
in  the  title  —  is  an  entire  chapter  treating  of  Underground  Con- 
ductors. This  has  been  written  with  the  kind  assistance  of 
Mr.  C.  J.  Beaver  who  has  not  only  furnished  most  of  the  data 
relating  to  underground  cables,  but  has  also  read  and  criticised 
the  matter  presented  in  Chapter  VII. 

A  portion  of  the  material  which  was  originally  contained  in  the 
Appendix  has  been  incorporated  in  the  text;  but  much  of  the  first 
edition  has  been  entirely  omitted,  either  because  its  inclusion 
is  no  longer  necessary  owing  to  the  rapid  strides  that  have  been 
made  of  late  years  in  the  general  knowledge  of  electrical  power 
transmission,  or  because  it  has  been  replaced  by  new  material 
believed  to  be  of  more  value  to  the  student  or  practical  engineer. 

The  chapter  describing  the  Thury  system  of  transmission  by 
continuous  currents  has  been  retained  with  only  slight  changes 
and  additions.  It  has  not  been  deemed  expedient  to  omit  this 
entirely  because,  although  few  American  engineers  have  taken 
the  trouble  to  familiarize  themselves  with  this  system  of  trans- 
mission, there  are  conditions  under  which  it  has  certain  in- 
disputable advantages  which  European  engineers  have  not  been 
slow  to  recognize. 

The  costs,  both  of  material  and  labor,  which  are  given  in 
Chapter  III,  are  not  representative  of  market  conditions  on  or 
about  the  date  of  publication  of  this  book.  Their  principal 
use  is  to  give  an  idea  of  the  relative  costs  of  different  parts  of  a 
transmission  system.  They  are  based  on  trade  conditions 
prevailing  during  the  two  or  three  years  immediately  preceding 
the  war.  A.  STILL. 

PURDUE  UNIVERSITY, 
LAFAYETTE,  INDIANA, 
June,  1919. 


PREFACE  TO  FIRST  EDITION 

Although  this  book  treats  mainly  of  the  fundamental  principles 
and  scientific  laws  which  determine  the  correct  design  of  over- 
head electric  transmission  lines,  it  has  been  written  primarily 
to  satisfy  the  needs  of  the  practical  engineer.  An  attempt  has 
been  made  to  give  the  reasons  of  things — to  explain  the  deriva- 
tion of  practical  methods  and  formulas — in  the  simplest  possible 
terms:  the  use  of  higher  mathematics  has  been  avoided;  but 
vector  diagrams,  supplemented  where  necessary  with  trigono- 
metrical formulas,  have  been  freely  used  for  the  solution  of  alter- 
nating-current problems.  It  is  therefore  hoped  that  the  book 
may  prove  useful,  not  only  to  the  practical  designer  of  trans- 
mission lines,  but  also  to  those  engineering  students  who  may 
wish  to  specialize  in  the  direction  of  Power  Generation  and 
Transmission,  for  these  will  find  herein  a  practical  application 
of  the  main  theoretical  principles  underlying  all  Electrical 
Engineering. 

The  subject  is  treated  less  from  the  standpoint  of  the  construc- 
tion engineer  in  charge  of  the  erection  work,  as  of  the  office 
engineer  whose  duty  it  is  to  make  the  necessary  calculations 
and  draw  up  the  specifications.  The  considerations  and  practi- 
cal details  of  special  interest  to  the  engineer  in  charge  of  the 
work  in  the  field  have  already  been  presented  in  admirable  form 
by  Mr.  R.  A.  Lundquist  in  his  book  on  Transmission  Line 
Construction. 

Much  of  what  appears  in  these  pages  is  reprinted  with  but 
little  alteration  from  articles  recently  contributed  by  the  writer 
to  technical  journals;  but  in  the  selection  and  co-ordination  of 
this  material,  the  scheme  and  purpose  of  the  book  have  steadily 
been  kept  in  mind. 

Systems  of  distribution,  whether  in  town  or  country,  are  not 
touched  upon:  the  subjects  dealt  with  cover  only  straight  long- 
distance overhead  transmission.  It  is  true  that,  when  treating 
of  lightning  protection,  it  is  the  machinery  in  the  station  buildings 
rather  than  the  line  itself  that  the  various  devices  referred  to  are 
intended  to  protect;  and,  when  considering  the  most  economical 


viii  PREFACE 

system  of  transmission  under  given  circumstances,  a  thorough 
knowledge  of  the  requirements  and  possibilities  in  the  arrange- 
ment of  generating  and  transforming  stations  is  assumed;  but 
these  engineering  aspects  of  a  complete  scheme  of  power  develop- 
ment are  not  included  in  the  scope  of  this  book. 

In  the  Appendix  will  be  found  reprints  of  some  articles  dealing 
with  theoretic  aspects  of  long-distance  transmission  which, 
although  believed  to  be  of  interest  to  anyone  engaged  on  the 
design  of  transmission  lines,  are  not  essential  to  the  scheme  of 
the  book.  In  the  Appendix  will  also  be  found  complete  speci- 
fications for  a  wood  pole  and  steel  tower  line  respectively:  these 
should  be  helpful,  not  so  much  as  models  for  other  specifications 
— every  engineer  is  at  liberty  to  draw  these  up  in  his  own  way — 
but  rather  as  containing  suggestions  and  reminders  that  may  be 
of  service  when  specifying  and  ordering  materials  for  an  actual 
overhead  transmission. 

The  writer  desires  to  thank  the  editors  of  the  following  tech- 
nical journals  for  permission  to  reprint  articles  or  portions  of 
articles  which  they  have  published  from  time  to  time:  Electrical 
World,  New  York;  Electrical  Times,  London;  Canadian  Engineer, 
Toronto;  Western  Engineering,  San  Francisco;  Journal  oj  Elec- 
tricity, Power,  and  Gas,  San  Francisco. 

PURDUE  UNIVERSITY, 
LAFAYETTE,  INDIANA, 
August,  1913. 


CONTENTS 

PAGE 

Preface  to  Second  Edition v 

Preface  to  First  Edition vii 

List  of  Symbols xv 

CHAPTER  I 

Introductory  and  General 1 

CHAPTER  II 
Electrical  Principles  and  Theory — Elementary 

1.  Losses  in  Transmission 13 

2.  Transmission  by  Continuous  Currents 14 

3.  Transmission  by  Single-phase  Alternating  Currents 14 

4.  Transmission  by  Two-phase  Currents 15 

6.  Transmission  by  Three-phase  Currents 17 

6.  Relative  Cost  of  Conductors  Required  on  the  Various  Systems. . .  19 

7.  Grounding  the  Neutral  on  High-tension  Overhead  Transmissions . .  23 

8.  Regulation:  Effect  of  Line  Inductance  on  the  Transmission  of  Al- 

ternating  Currents 24 

9.  Fundamental   Vector  Diagram  for  Line  Calculations:  Capacity 

Neglected 25 

10.  Effect  of  Capacity  on  Regulation  and  Line  Losses 28 

11.  Use  of  Fundamental  Diagram  for  Three-phase  Calculations 32 

CHAPTER  III 
Economic  Principles  and  Calculations 

12.  Introductory • 36 

13.  Choice  of  System 37 

14.  Type  of  Transmission  Line 37 

15.  Length  of  Span 38 

16.  Effect  of  Span  Variations  on  Cost  of  Steel  Towers 39 

17.  Cost  of  Wood  Poles 41 

18.  Cost  of  Insulators 42 

19.  Duplicate   Lines 42 

20.  Costs  of  Typical  Transmission  Lines 43 

21.  Cost  of  Overhead  Conductors 48 

22.  Economic  Size  of  Conductor.     Kelvin's  Law 49 

ix 


x  CONTENTS 

PAGE 

23.  Practical  Method  of  Applying  Kelvin's  Law 51 

24.  Economic  Ohmic  Voltage  Drop 52 

25.  Economic  Voltage,  and  Calculation  of  Conductor  Sizes 53 

26.  Example   Illustrating  Quick  Method  of  Determining  Economic 

Size  of  Conductors 55 

27.  Estimation  of  Amount  and  Cost  of  Energy  Wasted  in  Conductors.  57 

28.  Estimation  of  Percentage  to  Cover  Annual  Interest  and  Depre- 

ciation on  Conductors 58 

29.  Economic  Voltage 59 

30.  Costs  Other  than  Transmission  Line,  Liable  to  be  Influenced  by 

Voltage  Variations 60 

31.  Annual  Charges  Depending  on  Voltage 60 

32.  Depreciation 61 

33.  Example:  Method  of  Determining  Most  Economical  Voltage 63 

34.  Closer  Estimate  of  Economical  Voltage 66 

CHAPTER  IV 
Electrical  Principles  and  Calculations 

36.  Materials 69 

36.  Copper 70 

37.  Aluminum 70 

38.  Iron  and  Steel 71 

39.  Copper-clad  Steel 72 

40.  Stranded  Cables  with  Steel  Wire  Core 73 

41.  Physical  Constants  and  Sizes  of  Commercial  Conductors 74 

42.  Skin  Effect 77 

43.  Inductance  of  Transmission  Lines 79 

44.  Effect  of  Taking  into  Account  the  Return  Conductor. 80 

45.  Effect  of  Flux  Lines  in  the  Material  of  the  Conductor 82 

46.  Iron  as  a  Material  for  Transmission  Line  Conductors 83 

47.  Apparent  Resistance  of  Iron  and  Steel  Conductors 84 

48.  Internal  Reactance  of  Iron  and  Steel  Conductors 85 

49.  Example  of  Calculations  of  Iron  Wire  Conductors 88 

60.  Inherent  Regulation  of  Transmission  Line.     Regulation  Diagrams .  89 

61.  Pressure  Available  at  Intermediate  Points  on  a  Transmission  Line. .  94 

62.  Capacity  of  Transmission  Lines 96 

53.  Capacity  of  Three-phase  Lines 97 

64.  Charging  Current  Due  to  Capacity  of  Transmission  Lines 99 

56.  Effect  of  Distributed  Capacity  and  Inductance .101 

56.  Electrical  Calculation  of  Lines  with  Appreciable  Capacity 102 

67.  Numerical  Examples  Illustrating  Use  of  Formulas  for  the  Cal- 

culation of  Power  Factor  and  Voltage  Drop 104 

68.  Distinction  Between  Regulation  and  Line  Drop 108 

69.  Line  Losses 110 

60.  Control  of  Voltage  on  Transmission  Lines 112 

61.  Effect  of  Boosting  Voltage  at  Intervals  along  a  Transmission  Line.  114 


CONTENTS  xi 

PAGE 

62.  Control  of  Power  Factor. 115 

63.  Use  of  Rotary  Reactors  to  Control  the  Voltage 117 

64.  Power  Factor  of  Load 120 

65.  Grounded  versus  Isolated  Transmission  Systems 120 

66.  Interference  between  Power  and  Telephone  Lines 122 

67.  Insulation  of  Telephone  Lines 122 

68.  Electrostatic  Induction 122 

69.  Magnetic  Induction 123 

70.  Fault  Localizing 125 

CHAPTER  V 
Insulation  of  Overhead  Transmission  Lines 

71.  Insulator  Materials 127 

72.  Design  of  Insulators 129 

73.  Pin-type  Insulators 131 

74.  Suspension-type  Insulators 134 

75.  Wall  and  Roof  Outlets 143 

76.  Design  of  Insulating  Bushings 146 

77.  Condenser  Type  of  Bushing 150 

78.  Formation  of  Corona,  and  Accompanying  Losses  of  Power 151 

79.  Corona  Considered  as  "Safety  Valve"  for  relief  of  High-frequency 

Surges  or  Over-voltage  Due  to  Any  Cause 156 

80.  Spacing  of  Overhead  Conductors 156 

81.  Practical  Limitations  of  Overhead  Transmission-line  Voltages 158 

82.  Factors  of  Safety:  Rating  and  Testing  of  Line  Insulators 159 

CHAPTER  VI 
Protection  against  Lightning — Transient  Phenomena 

83.  Theoretical  Considerations 163 

84.  Frequency  of  Oscillations 167 

85.  Wave  Length 168 

86.  Reflection  of  Travelling  Waves 170 

87.  Line  Disturbances  Caused  by  Switching  Operations 173 

88.  Lightning 174 

89.  Protection  of  Overhead  Systems  against  Direct  Lightning  Strokes 

and  Sudden  Accumulations  of  High  Potential  Static  Charges.   175 

90.  Protection  of  Insulators  from  Power   Arcs 177 

91.  Methods  of  Grounding 178 

92.  Relieving   Conductors   of   High  Potential    "Static."     Water  Jet 

Arresters 179 

93.  Horn  Gap 180 

94.  Multiple-gap  Low  Equivalent  Arrester 183 

95.  Spark-gap  Arresters  with  Circuit  Breakers  or  Re-setting  Fuses. . . .  186 

96.  Aluminum  Cell  Arrester T .  187 


xii  CONTENTS 

PAGE 

97.  Condensers 189 

98.  Spacing  of  Lightning  Arresters 192 

99.  Choke  Coils 192 

100.  Arcing  Ground  Suppressor 194 

101.  General  Remarks  on  Lightning  Protection 195 

CHAPTER  VII 
Transmission  of  Energy  by  Underground  Cables 

102.  Introductory 199 

103.  Submarine  Power  Cables 200 

104.  Voltage  Limitations  of  Underground  Cables 201 

105.  Types  and  Construction  of  Power  Cables 202 

106.  Methods  of  Laying  Underground  Cables 206 

107.  Costs  of  Underground  and  Transmission  Lines 208 

108.  Cable  Terminals.     Junction  with  Overhead  Lines 211 

109.  Design  of  Cables 212 

110.  Economical  Core  Diameter  of  High-pressure  Cables 214 

111.  Grading  of  Cables 216 

112.  Three-core  Cables 217 

113.  Capacity  and  Charging  Current  of  Three-core  Cables 217 

114.  Example   of    Design   of   Single-phase  Concentric  E.H.T.   Power 

Cable 221 

115.  Losses  in  Underground  Cables 223 

116.  Temperature  Rise  of  Insulated  Cables 225 

117.  Reliability  of  Cable  Systems.     Joints;  Electrolysis 228 

CHAPTER  VIII 
Transmission  of  Energy  by  Continuous  Currents 

118.  General  Description  of  the  Thury  System 233 

119.  Straight  Long-distance  Transmission  by  Continuous  Currents...  236 

120.  Insulation  of  Line  when  Carrying  Continuous  Currents 237 

121.  Relative    Cost  of  Conductors:  Continuous  Current  and  Three- 

phase  Transmissions 238 

122.  Concluding  Remarks  on  Direct-Current  Transmission 242 

CHAPTER  IX 
Mechanical  Principles  and  Calculations — Overhead  Conductors 

123.  Introductory 248 

124.  Graphical   Statics   Applied  to   Transmission-line   Calculations — 

General  Problem .  .  249 

125.  Stretched  Wire.     Supports  on  Same  Level 250 

126.  Supports  at  Different  Elevations , 254 


CONTENTS  xiii 

PAGE 

127.  Calculation  of  Sag  with  Supports  on  an  Incline 258 

128.  Example  Illustrating  Use  of  Formulas 259 

129.  Conclusions — Overhead  Lines  on  Steep  Grade 261 

130.  Effect  of  Temperature  Variations  on  Overhead  Wires 261 

131.  Abnormal  Stresses  in  Wires  due  to  Wind  and  Ice 264 

132.  Swaying  of  Wires  in  Strong  Wind 273 

133.  Calculation  of  Total  Stress  in  Overhead  Wires 273 

134.  Effect  of  Temperature  Variations  on  Sag  and  Stress 278 

135.  Calculations  of  Sags  and  Tensions  under  any  Conditions  of  Load 

and  Temperature 281 

136.  Tensions  in  Conductors  when  Spans  are  of  Different  Lengths 288 

137.  Tension  in  Different  Sized  Wires  on  the  Same  Span. 288 

138.  Further  Example  Illustrating  Temperature-sag  Calculations 289 

139.  Sag-temperature     Calculations     with     Supports      at     Different 

Elevations 290 

140.  Length    of     Spans:  Conductor     Materials:  Copper:  Aluminum: 

Iron 298 

141.  Factors  of  Safety:  Joints  and  Ties 301 


CHAPTER  X 
Transmission  Line  Supports 

142.  General  Considerations:  Types  of  Transmission  Line  Supports. . .  303 

143.  Wood  Pole  Lines:  Kinds  of  Wood  Available 305 

144.  Typical  Woodpole  Lines 306 

145.  Life  of  Wood  Poles:  Preservative  Treatment 307 

146.  Insulating  Qualities  of  Wood  Poles 313 

147.  Weight  of  Wood  Poles 313 

148.  Strength  and  Elasticity  of  Wood  Poles 314 

149.  Calculation  of  Pole  Strengths 316 

150.  Deflection  of  Wood  Poles. 318 

151.  Calculation  of  Pole  Deflections 318 

152.  Pole  Foundations 320 

153.  Spacing  of  Poles  at  Corners:  Guy  Wires 321 

154.  Load  to  be  Carried  by  Corner  Poles 322 

155.  Props  or  Struts:  Wood  Poles  in  Compression 323 

156.  Reinforced  Concrete  Poles 324 

157.  Weight  and  Cost  of  Concrete  Poles 325 

158.  Strength  and  Stiffness  of  Concrete  Poles 327 

159.  Steel  Poles  and  Towers:  Introductory  Remarks 329 

160.  Flexible  Towers 331 

161.  Steel  Poles  for  Small  Short-distance  Transmission  Schemes 334 

162.  Loads  to  be  Resisted  by  Towers 335 

163.  Design  of  Steel  Towers 336 

164.  Stresses  in  Compression  Members  of  Tower  Structures 337 

165.  Outline  of  Usual  Procedure  for  Calculating  Stresses  in  Tower 

Members. . .  340 


xiv  CONTENTS 

PAGE 

166.  Stiffness  of  Steel  Towers:  Deflection  Under  Load 344 

167.  Tower  Foundations 344 

168.  Concluding  Remarks  Regarding  Steel  Tower  Design 347 

169.  Determining  Position  of  Supports  on  Uneven  Ground 348 

170.  Study  of  Deflections  and  Stresses  in  Flexible  Tower  Lines 349 

171.  Numerical  Exam  pie:  Transmission  Line  with  Flexible  Supports.  .  352 

172.  Erection  of  Steel  Tower  Transmission  Lines 359 

APPENDIX  1 

INDUCTANCE   OF   TRANSMISSION  LINES  WITH  ANY  ARRANGEMENT  OF 

PARALLEL  CONDUCTORS 361 

Calculation  of  Total  Resultant  Flux  Surrounding  one  Conductor 
when  there  are  Several  Return  Conductors — Calculation  of  E.m.f. 
of  Self  and  Mutual  Induction — Numerical  Example:  Three-phase 
Transmission — Inductance  of  Electric  Transmission  Lines  as 
Affected  by  the  Subdivision  of  the  Circuits  and  the  Arrangement 
of  the  Conductors. 

APPENDIX  II 

SPECIFICATION  FOR  WOOD  POLE  TRANSMISSION  LINE 375 

General  Description  of  Transmission  Line — Clearing — Poles — 
Cross-arms — Grading — Pole  Setting — Grounding — Spans — Angles 
and  Curves — Guying — Insulators — Stringing  Wires — Locating  and 
Numbering  Poles. 

APPENDIX  III 

SPECIFICATIONS  FOR  STEEL  TOWER  TRANSMISSION  LINE 386 

General  Description  of  Line — Duties  of  Engineer  in  Charge  of 
Construction — Clearing — Towers — Foundations  for  Towers — 
Grounding — Guying — Angles — Erection  of  Towers — Insulation — 
Conductors — Joints  in  Conductors — Spans  and  Wire  Stringing 
—Specification  for  Steel  Towers— Specification  for  Porcelain 
Insulators. 

INDEX. . .  . .  397 


LIST  OF  SYMBOLS 


A  =  area  of  cross  section. 

a  =  temperature-elongation  coefficient. 

a  =  percentage  to  cover  annual  interest  and  depreciation. 

a  =  n~  in  capacity  formulas. 

B  =  magnetic  flux  density  (gauss). 
B.  &  S.  =  Brown  and  Sharpe  wire  gauge. 

6  =  barometric  pressure:  cm.  or  inches  of  mercury. 
C  =  electrostatic  capacity  (farad). 

Cm  =  electrostatic  capacity;  microfarads  (usually  per  mile  of  con- 
ductor). 
Ce  =  effective  equivalent  capacity,  core  to  neutral,  of  underground 

cables. 

D  =  flux  density  in  electrostatic  field  (coulombs  per  sq.  cm.). 
D  =  butt  diameter  of  wood  pole. 
d  =  distance  between  centers  of  parallel  wires. 
d  =  diameter  of  wood  pole. 
dg  =  diameter  of  wood  pole  at  ground  level. 
E  =  electromotive  force  (e.m.f.):  difference  of  potential  (volts). 
E  =  voltage  between  lines  at  receiving  end. 
Ea  =  voltage,  line  to  ground,  on  A.  C.  system  (in  comparison  with 

D.  C.). 

En  —  voltage  between  wire  and  neutral  (usually  at  receiving  end). 
E0  =  disruptive  critical  voltage  (corona) — (r.m.s.  value  of  sine  wave). 
Er  =  "economic"  ohmic  drop,  in  volts  per  mile  of  conductor. 
Et  =  visual  critical  voltage  (corona). 

e  =  e.m.f. — usually  volts, 
e.m.f.  =  electromotive  force. 

F  =  wind  pressure:  Ib.  per  sq.  ft. 

/  =  frequency  (number  of  periods  per  second). 

G  =  potential  gradient  =  -^r  (volts  per  centimeter). 

<j»  =  potential  gradient  at  which  corona  begins  to  appear. 
H  =  height  of  pole  or  tower  (feet  or  inches;  refer  to  text). 
H  =  intensity  of  magnetic  field;  magnetizing  force  (gilbert  per 

centimeter;  or  gauss). 

h  =  depth  of  footing  below  ground  level  (Art.  167) — feet. 
h  =  difference  in  level  between  two  points  of  support  of  overhead 

wire  (feet). 
hp.  =  horsepower. 

/  =  current  (amperes). 


LIST  OF  SYMBOLS 

I  =  moment  of  inertia  of  pole  section  (Art.  151). 
Ia  =  current  in  three-phase  line  (in  comparison  with  D.  C.). 
/«.  =  capacity — or  charging — current  (amperes). 
/«  =  total  line  current  at  a  given  point  when  current  varies;  on 

account  of  distributed  capacity. 
i  =  current — usually  in  amperes. 

K  =  a  numerical  constant  (in  capacity  formulas  K  =  8.84  X  10~14). 
k  =  a  numerical  constant  (steel  tower  design;  Art.  164). 
k  =  "skin  effect"  factor. 
k  =  specific  inductive  capacity;  dielectric  constant;  permittivity. 

k  =  •£-!•  =  1.5  times  the  wt.  in  Ib.  of  a   cubic  inch  of  conductor 

oA 

material. 

k.v.a.  =  kilovolt-amperes. 
k.w.  =  kilowatts. 

L  =  distance  of  transmission  (miles). 
L  =  inductance,  or  flux-linkages  per  unit  current  (henry). 
Li  =  internal  inductance  of  straight  conductor. 
I  =  a  length  =  length  of  dielectric  flux  line  (cm.). 
I  =  length  of  span — horizontal  distance  between  supports  (feet). 
I  =  length  of  wire  or  cable  (cm.). 
I  =  length  of  wood  strut  and  of  compression  members  of  steel 

towers  (inch). 
I'  =  straight  line  distance  between  supports  of  wire  span. 

\A   1  refer  to  Fig.  93. 
IB   } 

M  =  modulus  of  elasticity-ratio 

(M)  =  circular  mils  per  ampere. 

mutual  capacity    . 

m  =  ratio rr —  -^  in  formulas  for  suspension  insulators. 

capacity  to  ground 

(m)  =  circular  mils. 

correction  factors  in  corona  formulas. 

m.m.f.  =  magnetomotive  force  (gilbert). 

n  =  number  of  phases  or  conductors  of  a  polyphase  system. 
n  =  number  of  units  in  a  string  of  suspension  insulators. 
=       .          resultant  loading  per  foot  of  wire 

'10  loading  per  foot  due  to  wt.  of  wire  only' 
P  =  power. 
P  =  permeance. 
P  =  pull;  tension;  force;  (Ib.). 

PA  =  horizontal  component  of  total  tension  in  suspended  wire. 
p  =  wind  pressure  per  foot  length  of  wire  (Ib.). 
p  =  price  of  100  Ib.  of  transmission  line  conductor. 
Pi  =  cost  per  kilowatt-year  of  I2R  losses. 
Q  =  quantity  of  electricity. 
R  =  inside  radius  of  metal  cylinder  surrounding  a  conductor. 


LIST  OF  SYMBOLS  xvii 

R  =  ground-level  radius  of  cone  of  earth  to  be  lifted  by  tower  leg 

(ft-)- 

R  =  resistance    (ohm):    Resistance   of   one  conductor  of  a  trans- 
mission line.     Resistance  per  mile  of  one  conductor. 
R  =  reluctance  (oersted). 
Ra  —  non-inductive  resistance  in  ground  connection  from  lightning 

arrester. 
Ra  =  resistance  per  mile  of  wire  (three-phase,  in  comparison  with 

D.  C.  system). 

Ri  =  insulation  resistance  (megohms  of  one  mile  of  cable). 
Rp  =  joint  resistance  of  all  conductors  of  a  transmission  line  con- 
nected in  parallel. 

r  =  radius  or  semi-diameter  of  cylindrical  conductor. 
r  =  least  radius  of  gyration  (inches). 
S  =  stress  (Ib.  per  square  inch), 
s  =  vertical  sag  (feet), 
s'  =  maximum  deflection  of  wire  from  straight  line  joining  points 

of  support  (line  on  an  incline). 
Sc  =  stress  at  "critical"  temperature. 
sc  =  sag  at  "critical"  temperature. 
Sm  =  maximum  stress. 
T  =  temperature  rise  (degrees  Centigrade). 
t  =  temperature  (degrees  Centigrade  or  Fahrenheit;  refer  text). 
I  =  interval  of  time  (usually  seconds). 
t  =  constant  defining  taper  of  wood  poles  (Art.  147). 
tc  =  "critical"  temperature — degrees  Fahrenheit  (sag-temperature 

calculations). 

V  =  velocity  of  wind  (miles  per  hour). 
V  =  volume  of  frustrum  of  cone  (cubic  feet). 
V  =  volts  between  lines  at  generating  end. 
Ft  =  reactive  voltage  drop  due  to  "internal  reactance"  only. 
F«  =  volts  between  wire  and  neutral  at  generating  end. 
W  =  power  (watts). 
W  =  resultant  pull  on  corner  pole  (Ib.). 
w  =  total  I2R  loss  in  conductors  of  a  transmission  line. 
w  =  weight  per  foot  length  of  overhead  wire  (Ib.). 
wr  =  resultant  or  total  load  on  wire  (Ib.  per  foot). 
X  =  reactance  (ohms). 
Z  =  impedance  (ohms). 
Z  =  section  modulus;  being  ratio 

moment  of  inertia  of  section 

distance  of  center  of  gravity  from  edge  of  section ' 
A  =  current  density  (amperes  per  square  inch). 
5  =  air  density  factor  (corona  formulas). 

5  =  deflection  at  top  of  pole  or  tower  (inch). 

0  -  an  angle:  cos  9  =  power  factor  of  load  (usually  at  receiving  end 
of  line). 

6  =  angle  between  direction  of  transmission  line  and  horizontal  line 

in  the  same  vertical  plane. 


iii  LIST  OF  SYMBOLS 

0  =  angle  of  natural  slope  of  earth. 

X  =  length  of  wire  between  two  points  of  support  (feet). 
Xc  =  length  of  wire  at  "critical  temperature"  in  still  air. 
\e  =  change  in  length  of  overhead  wire  (feet). 
fj,  =  magnetic  permeability  =  B/H . 
v  =  3.1416  (approximately). 

p  =  resistivity,  or  specific  resistance  (megohms  per  centimeter  cube). 
$  =  magnetic  flux  (maxwell). 

<P  =  an  angle ;  usually  the  power  factor  angle  at  sending  end  of  line, 
cos  <p  =  power  factor  of  insulated  cables  on  open  circuit. 
¥  =  dielectric  flux  (coulomb). 
«  =  2ir/. 


ELECTEE  POWEE  TEANSMISS10N 

PRINCIPLES  AND  CALCULATIONS 


CHAPTER  I 
INTRODUCTORY  AND  GENERAL 

Energy  can  be  transmitted  electrically  by  conductors  placed 
either  above  or  below  ground.  The  cost  of  a  system  of  under- 
ground insulated  cables  is  always  higher  than  that  of  an  equiva- 
lent overhead  transmission;  but  there  are  conditions,  especially 
in  Europe,  under  which  overhead  wires  are  not  desirable  or  per- 
missible, and  the  whole  or  a  portion  of  the  transmission  line  must 
then  be  placed  underground. 

The  electrical  transmission  of  energy  over  long  distances  at 
high  pressures  must  necessarily  be  by  overhead  conductors,  and 
even  at  the  lower  pressures — up  to  about  45,000  volts,  beyond 
which  underground  cables  would  not  be  suitable — by  far  the 
greater  number  of  transmission  lines  are  overhead.  It  follows 
that  this  book  is  mainly  concerned  with  the  problems  of  overhead 
electric  power  transmission;  but  since  underground  cables  may 
have  to  be  used  in  certain  sections  of  a  proposed  transmission 
scheme,  their  characteristics,  uses,  and  limitations,  will  be  dis- 
cussed in  Chapter  VTI. 

An  overhead  electric  power  transmission  line,  consisting  as  it 
does  of  wires  stretched  between  insulators  on  poles  or  structures 
the  main  purpose  of  which  is  to  maintain  the  conductors  at  a 
proper  distance  above  the  ground  level,  may  appear  at  first  sight 
to  be  a  very  simple  piece  of  engineering  work.  It  is  indeed  true 
that  the  erection  of  an  overhead  line  of  moderate  length,  capable 
of  giving  good  service  on  a  comparatively  low-pressure  system, 
does  not  present  any  insurmountable  difficulties  to  a  man  of 
ordinary  engineering  ability;  but  whether  or  not  such  a  line  will 
be  the  best  possible  line  for  the  particular  duty  required  of  it, 


2  ELECTRIC  POWER  TRANSMISSION 

depends  very  much  upon  the  knowledge,  skill,  and  experience  of 
the  designer.  By  the  best  line  should  be  understood  a  line 
which  is  not  only  substantially  and  lastingly  constructed,  but  in 
connection  with  which  economic  considerations  have  not  been 
overlooked. 

It  is  an  easy  matter  to  design  a  bridge  of  ample  strength  for  the 
load  it  has  to  carry,  or  a  transmission  line  with  conductors  of  so 
large  a  size,  insulators  with  so  large  a  factor  of  safety,  and  supports 
so  closely  spaced  and  strong,  that  the  electrical  losses  will  be 
small  and  the  risk  of  mechanical  failure  almost  nil;  but  neither 
the  bridge  nor  the  transmission  line  will  reflect  credit  on  the  de- 
signing engineer  unless  he  has  had  before  him  constantly  the 
commercial  aspect  of  the  work  entrusted  to  him,  and  has  so 
chosen  or  designed  the  various  parts,  and  combined  these  in  the 
completed  whole,  that  all  economic  requirements  are  as  nearly 
as  possible  fulfilled. 

In  the  construction  of  electrical  plant  and  machinery,  such  as 
generators,  transformers,  and  switching  apparatus,  the  economic 
conditions  are,  as  it  were,  automatically  fulfilled,  owing  to  the 
competition  between  manufacturers,  each  one  of  which  is  a  spe- 
cialist in  his  own  particular  line  of  business.  This  competition, 
it  should  be  observed,  is  not  merely  in  the  matter  of  works  cost  or 
selling  price,  but  in  works  cost  plus  efficiency  and  durability. 
It  is  not  necessarily  the  cheapest  nor  the  most  costly  manufac- 
tured article  that  wins  in  the  long  run,  but  the  one  which  is 
commercially  best  suited  to  the  needs  of  the  user. 

In  the  lay-out  of  power  plants;  in  the  development  of  natural 
power  resources  and  the  transmission  of  electric  energy  from 
water  falls  or  coal  fields  to  the  industrial  centers,  the  engineer, 
who  may  or  may  not  be  influenced  by  possibly  conflicting  finan- 
cial interests,  has  much  scope  for  the  reckless  and  unwise  expen- 
diture of  other  people's  money.  He  must  resist  this  temptation 
— if  temptation  it  be — and  devote  himself  to  the  careful  study  of 
all  engineering  problems  from  the  economic  standpoint. 

The  cost  per  mile  of  a  finished  transmission,  whether  by  over- 
head or  underground  conductors,  is  not  all-important.  It 
may  frequently  be  said  to  be  of  importance  only  in  so  far  as  it 
influences  the  annual  cost  of  the  line,  which  annual  cost  is  under- 
stood to  include  interest  on  the  capital  sum  expended  on  the  line. 
If  a  heavy  section  of  copper  is  used  for  the  conductors,  the  loss 
of  energy  in  overcoming  resistance  will  be  less  than  with  a  lighter 


INTRODUCTORY  AND  GENERAL  3 

section,  but  the  initial  cost  will  be  greater:  there  is  only  one  par- 
ticular size  of  conductor  which  is  economically  the  right  size 
for  any  given  line  operating  under  definite  conditions,  and  this 
is  by  no  means  easy  to  determine  notwithstanding  the  apparent 
simplicity  of  what  is  usually  referred  to  as  Kelvin's  Law. 

Efficiency  of  service,  which  includes  reasonably  good  voltage 
regulation  and  freedom  from  interruptions,  must  necessarily  be 
merged  into  the  all-important  question  of  cost.  By  duplicating 
an  overhead  transmission  line  and  providing  two  separate  pole 
lines,  preferably  on  different  and  widely  separated  rights  of  way, 
insurance  is  provided  against  interruption  of  service  over  an 
extended  period  of  time;  but  whether  or  not  such  duplicate  lines 
shall  be  erected  must  be  decided  on  purely  economic  grounds. 

Again,  lightning  arresters  may  be  provided  in  abundance  at 
frequent  intervals  along  an  overhead  transmission  line,  and 
assuming — what  is  not  necessarily  the  case — that  such  profusion 
of  protective  devices  will  prevent  interruptions  which  are  other- 
wise liable  to  occur  through  lightning  disturbances,  it  does  not 
follow  that  they  should  be  installed.  Examples  of  this  kind  can 
be  cited  to  an  almost  unlimited  extent,  and,  in  Chapter  III 
which  deals  especially  with  economics,  an  attempt  will  be  made  to 
indicate  a  mode  of  procedure  in  designing  a  transmission  line 
from  this,  the  only  standpoint  of  importance  to  the  engineer;  but 
the  question  is  a  large  one  which  cannot  adequately  be  dealt  with 
by  set  rules  or  formulas.  In  such  cases  as  the  design  of  support- 
ing structures,  when  the  calculations  for  strength  have  been  made, 
it  is  the  designer  of  the  transmission  line  and  not  the  manufacturer 
of  the  steel  tower  who  should  decide  upon  the  factor  of  safety  to  be 
used,  for  this  is  the  prerogative  of  the  man  who  is  going  to  be 
held  responsible  for  the  commercial  success  of  the  undertaking. 
If  he  is  incompetent  or  timid,  he  will  allow  too  high  a  factor  of 
safety,  or  follow  blindly  in  the  footsteps  of  others  who  may  have 
been  equally  incompetent  or  timid.  If  he  is  sure  of  himself,  and 
has  carefully  checked  his  calculations  and  deductions,  he  may 
depart  from  precedent  and  construct  a  line  which  is  cheaper — 
not  only  in  appearance,  but  in  fact — that  any  line  previ- 
ously constructed  under  similar  conditions  and  within  the  same 
limitations. 

It  is  commonly  supposed  that  wood  poles  with  pin  type 
insulators,  spaced  something  less  than  200  feet  apart,  should  be 
used  on  small-power  lines  working  at  low  or  medium  pressures: 


4  ELECTRIC  POWER  TRANSMISSION 

yet  there  are  conditions  under  which  light  steel  "A"  frames, 
or  latticed  poles,  may  with  advantage  replace  wooden  poles.  In 
some  cases  taller  steel  structures  with  long  spans  might  even 
prove  to  be  the  better  construction  from  the  economical  stand- 
point. On  the  other  hand,  when  a  large  amount  of  power  has  to 
be  transmitted  over  long  distances  at  high  pressure — in  which 
case  the  suspension  type  of  insulator  is  almost  a  necessity — it 
does  not  follow  that  tall,  rigid,  wide-base,  steel  towers,  spaced 
about  eight  to  the  mile,  should  necessarily  be  adopted.  The 
so-called  "flexible"  steel  structures,  with  somewhat  closer 
spacing,  may  prove  more  desirable  for  several  reasons  apart  from 
the  costs  of  material  and  erection.  Even  wood  poles  may  have 
advantages  over  steel  structures,  especially  when  the  right  kind 
of  wood  pole  is  readily  obtainable,  and  when  the  cost  of  transport- 
ing the  steel  structures  from  factory  to  site  is  an  important 
item  of  the  total  cost.  The  engineer  should  be  slow  to  decide 
upon  the  kind  of  structure  and  average  length  of  span  best  suited 
to  the  character  of  the  country  through  which  the  line  will  be 
carried.  As  an  illustration  of  the  fact  that  no  particular  type 
of  construction  is  suitable  under  all  conditions,  it  is  interesting 
to  note  that,  of  the  important  electric  power  lines  in  all  parts  of 
the  world,  transmitting  energy  at  pressures  exceeding  50,000  volts, 
the  total  mileage  is  about  equally  divided  between  the  wood  pole 
and  steel  tower  types  of  construction.  A  good  example  of  wood 
poles  used  for  high-pressure  power  transmission  is  the  100,000- 
volt  three-phase  transmission  line  of  the  Montana  Power  Com- 
pany, where  each  supporting  structure  consists  of  two  45-foot 
cedar  poles  connected  at  the  top  by  a  single  cross-arm,  but  pro- 
vided with  no  additional  bracing  or  stiffening  members  between 
the  poles.  This  line  runs  from  Great  Falls  to  Deer  Lodge, 
Montana,  a  distance  of  140  miles;  it  consists  of  three  No.  4/0 
copper  conductors,  and  supplies  power  to  the  Chicago,  Milwaukee 
&  St.  Paul  Railway. 

The  climate  and  probable  weather  conditions  will  obviously 
have  an  important  bearing  on  the  safe  span  limit  and  mechanical 
design  of  the  line  generally.  The  effects  of  wind  and  ice  will 
be  referred  to  in  Chapter  IX. 

A  knowledge  of  the  country  through  which  an  overhead  trans- 
mission line  is  to  be  carried  is  essential  to  the  proper  design  of 
the  line  and  supporting  structures.  Without  a  knowledge  of 
the  natural  obstacles  to  be  reckoned  with,  including  the  direction 


INTRODUCTORY  AND  GENERAL  5 

and  probable  force  of  wind  storms,  and  whether  or  not  these 
may  occur  at  times  when  the  wires  are  coated  with  ice,  the  nature 
of  the  supports  and  the  economical  length  of  span  cannot  prop- 
erly be  determined.  On  the  Pacific  coast,  where  there  is  rarely, 
if  ever,  an  appreciable  deposit  of  sleet  on  overhead  conductors, 
it  is  possible  that  the  spacing  of  supports  may  generally  be  greater 
than  in  countries  where  the  climatic  conditions  are  less  favorable. 
At  the  same  time,  it  has  been  observed,  in  districts  where  the 
winters  are  severe  and  sleet  formation  on  conductors  of  frequent 
occurrence,  that  the  effects  of  storms  in  winter  on  wires  heavily 
weighted  with  ice,  and  offering  a  largely  increased  surface  to  the 
wind,  are  less  severe  than  in  summer  when  much  higher  wind 
velocities  are  sometimes  attained.  These  examples  are  here 
mentioned  to  emphasize  the  necessity  for  a  thorough  investiga- 
tion of  local  conditions  before  starting  upon  the  detailed  design 
of  a  proposed  transmission  line. 

There  are  obviously  many  preliminary  matters  to  be  considered 
and  dealt  with  before  the  actual  details  of  design  can  be  proceeded 
with;  but,  although  many  of  these  are  partly,  if  not  wholly, 
engineering  problems,  they  cannot  adequately  be  dealt  within 
the  limits  of  this  book,  or  indeed  within  the  limits  of  any  book, 
since  the  differences  in  local  conditions,  in  the  scope  and  commer- 
cial aim  or  end  of  a  transmission  system,  makes  it  next  to  im- 
possible to  formulate  rules  or  devise  methods  of  procedure  which 
can  be  of  general  utility. 

Assuming  that  it  is  proposed  to  transmit  energy  electrically 
from  a  point  where  the  power  can  be  cheaply  generated  to  an 
industrial  or  populous  center  where  there  is  a  demand  for  it,  a 
straight  line  drawn  on  the  map  between  these  two  points  will 
indicate  the  route  which,  with  possibly  slight  deviations  to  avoid 
great  differences  in  ground  level,  would  require  the  smallest 
amount  of  conductor  material  and  the  fewest  poles  or  supporting 
structures.  There  may  be  natural  obstacles  to  the  construction 
of  so  straight  a  line,  as  for  instance,  lakes  that  cannot  be  spanned, 
or  mountains  that  cannot  be  climbed;  but  even  the  shortest 
route  which  natural  conditions  would  render  possible  is  by  no 
means  necessarily  the  best  one  to  adopt.  The  right  of  way  for 
the  whole  or  part  of  the  proposed  line  may  have  to  be  purchased, 
and  the  cost  will  often  depend  upon  the  route  selected.  By 
making  a  detour  which  will  add  to  the  length  of  the  line,  it  may 
be  possible  to  avoid  crossing  privately  owned  lands  where  a  high 


6  ELECTRIC  POWER  TRANSMISSION 

annual  payment  may  be  demanded  for  the  right  to  erect  and 
maintain  poles  or  towers.  Again,  by  paralleling  railroads  or 
highways,  the  advantage  of  ease  of  access  for  construction  and 
maintenance  may  outweigh  the  disadvantage  of  increased  length. 
A  slightly  circuitous  route  may  take  the  transmission  line  near  to 
towns  or  districts  where  a  demand  for  power  may  be  expected 
in  the  near  future;  and  such  possibilities  should  be  taken  into 
account.  The  engineer  in  charge  of  the  preliminary  survey  work 
(a  section  of  transmission-line  engineering  which  is  not  dealt  with 
in  this  book)  should  bear  all  such  points  in  mind  and  compare  the 
possibilities  of  alternative  routes.  On  a  long  and  necessarily 
costly  transmission  line,  it  is  rarely  possible  to  spend  too  much 
time  and  thought  on  the  preliminary  work.  Money  so  spent  is 
usually  well  spent,  and  will  result  in  ultimate  economies. 

Coming  now  to  the  problems  of  a  more  strictly  engineering 
nature,  one  of  the  first  things  to  be  decided  upon  is  the  system 
of  electric  transmission,  whether  it  shall  be  by  continuous  cur-, 
rents  with  its  simple  two-wire  circuit  and  ideal  power  factor,  or 
by  single-  two-  or  three-phase  alternating  currents  with  manifold 
advantages  in  respect  to  pressure  transformations  and  adapta- 
bility for  use  with  commutatorless  motors,  but  handicapped  by 
low  power  factors  and  other  complications  due  to  the  inductance 
and  electrostatic  capacity  of  the  circuit. 

Although  nearly  all  long-distance  transmissions — especially 
on  the  continent  of  America — are  by  three-phase  currents,  the 
other  systems  will  be  referred  to  briefly  in  the  following  chapter; 
and  since,  with  the  latest  improvements  in  continuous  current 
machinery,  the  series  system  of  power  transmission  by  continuous 
currents  may,  under  favorable  conditions,  hold  its  own  in  this 
country  as  it  does  in  Europe,  an  entire  chapter  will  be  devoted 
to  a  discussion  of  the  points  for  and  against  the  use  of  continuous 
currents  on  long-distance  power  transmission  lines. 

The  question  of  underground  cables  versus  overhead  con- 
ductors does  not  give  rise  to  so  much  discussion  in  America  as  it 
does  in  Europe.  Apart  from  the  limitation  of  voltage,  which 
renders  underground  cables  unsuitable  for  long-distance  trans- 
mission, the  cost  of  an  underground  system  of  conductors  will 
generally  be  at  least  twice  that  of  the  equivalent  overhead  con- 
struction. The  difference  in  cost  is  more  noticeable  in  America 
where  reasonable  factors  of  safety  in  overhead  construction  have 
been  adopted,  than  in  Europe,  and  especially  in  England  where 


INTRODUCTORY  AND  GENERAL  7 

legislation  has  in  the  past  been  generally  unfavorable  to  the  devel- 
opment of  electrical  transmission.  Factors  of  safety  of  10  for 
wooden  poles,  6  for  steel  structures,  and  5  for  overhead  con- 
ductors, with  a  maximum  wind  pressure  of  30  Ib.  per  square 
foot,  are  not  likely  to  encourage  the  construction  of  overhead 
lines;  yet  these  figures  are  insisted  upon  by  the  British  Board  of 
Trade.  A  factor  of  safety  of  5  for  an  elastic  and  moderately 
flexible  system  of  overhead  wires  in  a  country  rarely  subject  to 
severe  sleet  and  wind  storms,  seems  to  have  been  selected  on 
purely  arbitrary  grounds.  So  high  a  figure  is  especially  objec- 
tionable in  that  it  involves  extra  tall  structures  or  uneconomically 
short  spans,  if  the  wires  are  to  clear  the  ground  at  a  reasonable 
height.  In  any  case,  overhead  conductors  hanging  in  festoons 
would  not  be  considered  good  engineering  practice  in  this  coun- 
try. From  the  artistic  point  of  view,  the  unsightly  appearance 
of  innumerable  poles  and  wires  does  not  appreciably  perturb  the 
average  American,  whereas  it  may  surely  be  said  that  a  conges- 
tion of  high-  and  low- voltage  circuits  as  seen  in  Fig.  I,1  would 
not  be  tolerated  in  the  vicinity  of  the  smallest  hamlet  in  England. 
These  reasons  account  for  the  more  frequent  use  of  high-tension 
underground  cables  in  Europe  than  in  America,  notwithstanding 
the  fact  that  interruptions  due  to  lightning  are  far  more  liable 
to  occur  here  than  there. 

The  choice  of  system  and  determination  of  the  most  economical 
transmission  voltage  involve  a  knowledge  of  the  cost  and  effi- 
ciency of  generating  and  transforming  machinery  and  controlling 
gear.  It  is  obvious  that  a  system  of  transmission  that  appears 
good  owing  to  the  low  cost  and  high  efficiency  of  the  line  itself, 
may  yet  be  unsuitable  and  uneconomical  because  of  the  high  cost 
or  unsatisfactory  nature  of  the  machinery  in  the  generating  and 
receiving  stations. 

Apart  from  capital  investment  and  power  efficiency,  a  factor 
of  the  greatest  importance,  almost  without  exception,  is  efficiency 
and  continuity  of  service.  At  the  present  time,  the  weakest 
link  in  a  power  system  with  long-distance  transmission  is  proba- 
bly the  line  itself.  Electrical  troubles  may  be  due  to  faulty  in- 
sulation, or  they  may  have  their  origin  in  lightning  or  switching 
operations  causing  high  frequency  oscillations  and  abnormally 
high  voltages,  leading  to  fracture  of  insulators  or  breakdown  of 
machinery.  Troubles  are  more  likely  to  be  due  to  mechanical 

1  Photograph  kindly  supplied  by  Messrs.  Archbold  Brady  and  Co. 


8  ELECTRIC  POWER  TRANSMISSION 

defects,  or  mechanical  injuries  sometimes  difficult  to  foresee  and 
guard  against.  Trees  may  fall  across  the  line,  landslides  may 
occur  and  overturn  supports,  or  severe  floods  may  wash  away 
pole  foundations;  and  against  such  possibilities  the  engineer  must, 
by  the  exercise  of  judgment  and  foresight,  endeavor  to  protect 
his  work.  The  width  of  the  right  of  way  should  depend  upon 
the  height  of  trees,  and  be  so  wide  that  the  tallest  tree  can- 
not fall  across  the  wires;  poles  and  towers  should,  if  possible,  be 
kept  away  from  the  sides  of  steep  hills  where  the  nature  of  the 
ground  suggests  the  possibility  of  falling  stones  or  of  landslides ; 
and,  in  regard  to  floods,  the  inhabitants  of  the  districts  through 
which  the  line  passes  are  usually  able  to  furnish  information  of 
use  in  indicating  where  trouble  from  this  cause  may  be  expected. 
Other  causes  of  mechanical  failure  are  storms  of  exceptional  vio- 
lence, either  with  or  without  a  heavy  coating  of  ice  on  the  con- 
ductors. When  strong  winds  blow  across  ice-coated  wires,  the 
danger  is  not  only  that  the  wires  themselves  may  break,  but  also 
that  the  resulting  horizontal  loading  of  the  poles  or  towers 
may  be  great  enough  to  break  or  overturn  them. 

Faulty  mechanical  design  of  the  line  as  a  whole,  and  improper 
supervision  or  inspection  during  construction,  will  account  for 
many  preventable  interruptions  to  service.  The  transmission 
line  considered  as  a  mechanical  structure  will  be  dealt  with  at 
some  length  in  Chapters  IX  and  X,  and  the  sample  specifica- 
tions for  a  wood  pole  and  steel  tower  line  respectively  (see  Appen- 
dices II  and  III),  together  with  detailed  material  schedules  with 
approximate  costs,  given  in  Chapter  III,  will  cover  some  practical 
details  which  should  be  helpful  when  designing  a  transmission 
line  to  operate  under  generally  similar  conditions. 

Although  the  electrical  and  mechanical  qualities  of  insulators 
and  conductor  materials  are  necessarily  somewhat  dependent 
upon  each  other,  an  attempt  will  be  made  to  deal  almost  exclu- 
sively with  electrical  calculations  and  the  electrical  character- 
istics of  transmission  lines  in  the  following  chapter,  and  again 
in  Chapters  IV,  V,  and  VI.  The  chapter  treating  particularly 
of  the  economic  aspect  of  transmission-line  design  will  follow 
immediately  after  Chapter  II,  because  it  is  well  to  determine 
provisionally  the  system  and  voltage  likely  to  be  most  suitable 
for  a  given  scheme,  before  entering  into  the  more  detailed  calcu- 
lations of  line  losses  and  regulation,  and  considering  the  practical 
requirements  in  matters  of  insulation  and  lightning  protection. 


FIG.  1. — Overhead  wires,  steel  towers,  and  poles. 

(Facing  Page  8) 


INTRODUCTORY  AND  GENERAL  9 

Before  closing  this  chapter  it  may  be  well  to  refer  to  a  few  mat- 
ters of  general  interest  that  are  not  dealt  with  in  succeeding 
chapters. 

Except  for  the  sample  specifications  in  the  Appendix,  pre- 
viously referred  to,  no  attempt  has  been  made  to  describe  com- 
plete transmission  lines,  with  details  of  construction  and 
operation;  but  the  more  purely  practical  details  of  construction 
have  already  been  admirably  presented  by  Mr.  R.  A.  Lundquist 
in  his  book  on  Transmission  Line  Construction,1  to  which  the 
reader  is  referred.  Descriptions  of  underground  cable  systems, 
together  with  a  complete  presentation  of  the  more  practical 
aspects  of  energy  transmission  by  underground  cables,  will 
be  found  in  Mr.  E.  B.  Meyer's  book  on  Underground  Trans- 
mission and  Distribution.1 

In  the  matter  of  crossing  highways  or  railroads  with  high- 
tension  conductors,  the  engineer  will  usually  have  to  abide  by  the 
rules  and  regulations  of  the  local  authority  or  railroad  company, 
however  unreasonable  or  unnecessary  the  particular  requirements 
may  be.  The  modern  tendency  is  to  avoid  a  multiplicity  of 
devices  intended  to  catch  falling  wires  or  ground  them  in  the 
event  of  a  breakage,  and  to  rely  mainly  on  short  spans  and 
exceptionally  sound  mechanical  construction  at  places  where 
the  falling  of  charged  wires  would  be  a  menace  to  life,  and  cause 
an  interruption  to  traffic. 

Reverting  again  to  the  all-important  matter  of  uninterrupted 
service,  it  is  obvious  that  reasonable  provision  should  be  made 
for  the  early  resumption  of  service  in  the  event  of  stoppage. 
When  two  parrallel  lines  are  run  all  the  way  between  generating 
and  receiving  stations,  it  is  usual  to  provide  section  switches 
and  by-pass  connections  about  every  20  miles,  at  the  points 
where  patrolmen  are  stationed.  Even  if  the  line  is  not  dupli- 
cated, it  is  usually  wise,  on  important  systems,  to  have  patrol- 
men's houses  every  15  or  20  miles,  depending  on  the  character 
of  the  country.  Emergency  houses,  containing  tools  and  sun- 
dry line  materials  such  as  wire  and  spare  insulators,  should  be 
provided  midway  between  the  patrol  houses. 

For  communication  between  generating  station  and  patrolmen 
and  substations,  a  telephone  circuit  is  almost  essential.  This  can 
be  run  on  the  same  poles  or  towers  as  the  high-tension  conductors; 
but,  if  possible,  it  should  be  run  on  separate  poles.  The  extra 

1  McGraw-Hill  Book  Co. 


10  ELECTRIC  POWER  TRANSMISSION 

cost  of  a  separate  pole  line  for  telephone  wires  is,  however,  the 
reason  for  these  wires  being  frequently  supported  on  the  same 
poles  as  the  transmission  wires.  This  leads  to  trouble  in  the 
case  of  a  ground  on  the  high-tension  wires  and,  indeed,  almost 
invariably  when  there  is  a  fault  on  the  power  system,  except,  of 
course,  when  the  power  service  is  entirely  interrupted.  Even 
when  carried  on  a  separate  pole  line,  the  telephone  circuit  is 
liable  to  be  useless  at  times  when  it  is  most  needed,  and  for  this 
reason  it  is  not  unusual,  on  important  lines,  to  provide  telegraphic 
instruments  in  addition  to  the  telephones,  and  men  that  are 
telegraph  operators  at  the  ends  of  the  line  and  also  at  any  inter- 
mediate points  where  switching  stations  may  be  provided. 

Closely  related  to  the  matter  of  continuous  and  efficient 
service  is  the  question  of  duplication  of  lines,  already  referred  to. 
This  has  to  be  decided  mainly  on  economic  grounds;  but,  at  the 
same  time,  the  purely  engineering  difficulties  may  become  very 
serious  if  an  attempt  is  made  to  transmit  very  large  amounts  of 
power  on  a  single  set  of  conductors.  It  is  not  possible  to  lay 
down  definite  rules  as  to  the  practical  limit  for  a  single  line; 
this  will  depend  on  the  distance  and  voltage,  and  therefore  on  the 
current  to  be  carried  per  conductor;  but  if  the  power  to  be  trans- 
mitted exceeds  30,000  kw.,  it  would  generally  be  wise  to  duplicate 
the  conductors,  even  if  carried  on  the  same  set  of  supporting 
towers;  and  if  the  total  power  exceeds  60,000  kw.,  two  separate 
tower  lines,  either  on  the  same  right-of-way  or  (preferably) 
following  different  routes,  will,  in  most  cases,  lead  to  ultimate 
economy. 

Among  recently  established  long-distance  high-voltage  trans- 
mission systems,  the  Big  Creek  to  Los  Angeles  line  of  the  Pacific 
Light  and  Power  Co.  is  one  of  the  most  important.  It  is  designed 
for  an  operating  pressure  of  150,000  volts  and  an  ultimate  ca- 
pacity of  300,000  kilowatts.  At  present  there  are  two  steel- 
tower  three-phase  lines,  running  parallel  over  a  total  distance  of 
240  miles,  with  an  average  distance  of  about  660  feet  between 
towers.  The  line  was  put  into  operation  in  1913.  Other  im- 
portant lines  are  those  of  the  Au  Sable  Electric  Co.  from  Battle 
Creek  to  Au  Sable,  Mich.  (245  miles),  and  of  the  Southern  Sierras 
Power  Co.  (238  miles),  both  operating  at  140,000  volts.  The 
latter  which  was  put  into  operation  in  1915,  departs  somewhat 
from  previous  practice  in  that  it  includes  many  outdoor  sub- 
stations. 


INTRODUCTORY  AND  GENERAL  11 

The  60,000-volt  system  of  the  Texas  Power  and  Light  Co. 
is  also  noticeable  for  its  outdoor  substations,  which  are  used 
exclusively  throughout  the  system.  The  present-day  tendency 
appears  to  be  toward  the  more  frequent  use  of  outdoor  substations 
where  not  only  the  section  disconnecting  switches,  but  also 
transformers,  lightning  arresters,  and  indeed  all  high-tension 
equipment,  are  exposed  to  the  weather.  The  control  of  local 
feeder  circuits  at  the  lower  pressures  would  be  by  means  of 
switch-panels  in  cheaply  constructed  houses  erected  near  the 
transformers. 

It  is  well  to  bear  in  mind  that  the  introduction  of  automatic 
switches  and  similar  devices  designed  to  save  labor  and  ensure 
the  rapid  changing  over  of  the  load  from  a  faulty  section  to  a 
sound  section  on  a  duplicated  transmission  line,  is  liable  to  lead 
to  unlooked-for  troubles;  and  even  the  generous  provision  of 
lightning  arresters,  especially  on  the  extra-high-tension  lines,  is 
not  necessarily  good  policy.  Simplicity,  and  the  avoidance  of 
unnecessary  joints,  rubbing  contacts  (as  in  switches  or  cut-outs), 
fuses  in  the  stations,  and  spark  gaps  or  arresters  along  the  line, 
should  generally  be  aimed  at;  but  there  will  always  be  exceptions 
to  such  rules. 

Careful  design  in  the  matters  of  material,  size  and  spacing  of 
conductors,  and  in  methods  of  support  and  insulation,  together 
with  scientific  selection  and  lay-out  of  poles  or  towers,  will  lead 
to  the  construction  of  transmission  lines  which  may  ultimately 
prove  to  be  the  strongest  instead  of  the  weakest  links  in  power 
transmission  schemes;  and  this  without  the  addition  of  more 
or  less  complicated  and  unreliable  automatic  and  so-called 
protective  devices,  and  at  a  cost  which  will  make  long-distance 
power  transmission  propositions  more  attractive  from  the  stock- 
holder's point  of  view  than  they  have  been  in  the  past. 

There  has  been  too  much  speculation  and  too  little  business 
connected  with  many  of  the  existing  power  transmission  under- 
takings, especially  where  water  power  is  available.  The  almost 
ideal  conditions  at  Niagara,  due  to  an  enormous  amount  of  energy 
being  available  in  the  proximity  of  industrial  cities  and  centers  of 
population,  are  probably  unique.  There  are  few,  if  any,  other 
water  power  sites  in  the  world  where  the  conditions  are  so  favor- 
able. If  competent  and  honest  engineers  were  always  employed 
by  promoters  of  water  power  developments,  for  electrical  trans- 
mission purposes,  and  if  the  reports  or  advice  of  such  experts  were 


12  ELECTRIC  POWER  TRANSMISSION 

acted  upon,  there  would  be  fewer  undertakings  carried  out  in  the 
wrong  manner.  In  determining  the  magnitude  of  any  proposed 
development,  and  the  distance  beyond  which  it  would  be  un- 
economical to  transmit  the  energy,  a  valuation  should  be  made  on 
the  basis  of  the  kilowatt-hours  per  year  available  for  which  there 
is,  or  is  likely  to  be,  a  market.  This  involves  experience,  together 
with  business  and  engineering  judgment  of  a  high  order,  on  the 
part  of  those  responsible  for  the  capital  invested;  and  although 
there  are  many  water  power  sites — especially  with  comparatively 
low  heads — which  still  await  development,  the  engineering  prob- 
lems in  connection  therewith  are  never  so  simple  as  to  render 
unnecessary  a  great  deal  of  investigation  and  careful  thought 
before  the  nature  of  the  development  and  the  best  manner  of 
carrying  out  the  work  can  be  properly  determined. 


CHAPTER  II 
ELECTRICAL  PRINCIPLES  AND  THEORY— ELEMENTARY 

The  purpose  of  an  electric  transmission  line  is  to  transmit 
energy  from  one  place  to  another;  and  it  is  the  engineer's  business 
to  design  and  construct  such  a  line  to  fulfil  its  purpose  in  the  best 
and  most  economical  manner. 

The  system  to  be  adopted  will  affect  the  design  of  the  gener- 
ating plant  and  of  the  motors  or  other  devices  through  which  the 
electric  energy  at  the  receiving  end  of  the  line  is  converted  for 
industrial  purposes  or  public  utility;  but,  in  this  chapter,  refer- 
ences to  alternative  systems  will  be  made  only  for  the  purpose  of 
comparing  them  in  the  matter  of  line  efficiency. 

1.  Losses  in  Transmission. — The  principal  cause  of  loss  of 
power  in  a  transmission  line  is  the  resistance  of  the  conductors. 
For  a  given  section  of  conductor,  the  power  dissipated  in  the  form 
of  heat  in  overcoming  the  ohmic  resistance  is  proportional  to 
the  square  of  the  current.  A  definite  amount  of  power  can  there- 
fore be  transmitted  with  less  loss  when  the  voltage  is  high  than 
when  it  is  low;  but,  on  each  particular  transmission,  there  is  a 
limit  to  the  pressure  beyond  which  there  is  nothing  to  be  gained 
in  the  matter  of  economy.  This  limit  is  determined  by  the  cost 
of  generating  and  transforming  apparatus  (which  will  be  greater 
for  the  higher  voltages),  by  the  greater  cost  of  insulators  and  of 
the  line  generally — owing  to  the  larger  spacing  required  between 
wires — and  also,  when  extra  high  pressures  are  reached,  by  the 
fact  that  the  power  dissipated  is  no  longer  confined  to  the  I2R 
losses  in  the  conductors,  but  occurs  also  in  the  form  of  leakage 
current  over  insulators,  and  in  the  air  surrounding  the  conductors. 
The  means  of  calculating  the  dielectric  losses  will  be  explained  in 
Chapter  V,  when  treating  of  corona  formation;  and  a  method 
for  determining  the  best  voltage  of  transmission  under  any 
given  conditions,  will  be  outlined  in  Chapter  III,  because  this 
is  essentially  an  economic  problem.  For  the  present  it  is  assumed 
that  a  total  amount  of  power  amounting  to  W  watts  has  to  be 
transmitted  over  conductors  of  known  resistance;  and  losses 
through  leakage  or  corona  will  be  considered  negligible. 

13 


14  ELECTRIC  POWER  TRANSMISSION 

2.  Transmission  by  Continuous  Currents.  —  If  E  is  the  voltage 
between  the  outgoing  and  return  wires  at  generating  end,  the 
current  is 

7_E 
E 
and  the  losses  in  transmission  are, 

2R  XP 

where  R  is  the  resistance  of  one  conductor  only.     The  loss  of 
pressure  per  wire  is 

R  X  I 

The  fact  that  continuous  currents  are  not  extensively  used  for 
the  transmission  of  power  to  a  distance  is  due  mainly  to  the 
difficulty  of  providing  sufficiently  high  pressures  to  render  such 
transmission  economical,  and  also  to  the  necessity  for  using  rotary 
machines  with  commutators  to  convert  the  transmitted  energy 
into  convenient  form  at  the  distant  end  of  the  line.  The  modern 
aspect  of  long-distance  transmission  by  means  of  continuous 
currents  will,  however,  be  dealt  with  at  some  length  in  Chapter 
VIII. 

3.  Transmission  by  Single-phase  Alternating  Currents. — The 
advantage  that  alternating  currents  have  over  direct  currents  is 
in  the  ease  with  which  pressure  transformations  can  be  effected 
be  means  of  static  converters.     On  a  constant-potential  system, 
the  distribution  of  power  in  scattered  districts,  at  any  voltage 
desired  by  the  consumer,  is  a  very  simple  matter. 

In  a  single-phase  two-wire  transmission,  the  conditions  would 
be  similar  to  those  of  a  direct-current  transmission  if  not  only  the 
load,  but  the  line  also,  could  be  considered  as  being  without 
inductance  or  electrostatic  capacity.  The  current  and  the  line 
losses  would  then  be  the  same  as  if  the  transmission  were  by  con- 
tinuous instead  of  alternating  currents. 

In  practice  the  inductance  must  always  be  reckoned  with  where 
alternating  currents  are  used;  this  inductance  is  not  only  that 
introduced  by  the  load  (usually  consisting  in  large  part  of  induc- 
tion motors),  but  is  partly  in  the  line  itself,  owing  to  the  loop 
formed  by  the  outgoing  and  return  conductors.  The  charging 
current  due  to  the  capacity  of  the  line  is  of  less  account  on  low- 
voltage  transmissions,  but  becomes  of  considerable  importance 
on  long  lines  working  at  high  pressures.  The  effects  of  induct- 
ance and  capacity  will  be  explained  later. 


ELECTRIC  PRINCIPLES  AND  THEORY  15 

Another  difference  between  alternating  and  continuous  cur- 
rents is  the  fact  that  an  alternating  current  has  the  effect  of 
apparently  increasing  the  resistance  of  the  conductor;  this  is 
due  to  the  uneven  distribution  of  the  current  over  the  cross- 
section  of  the  conductor.  A  small  percentage  of  the  alternating 
flux  of  induction  is  in  the  material  of  the  conductor  itself,  and 
this  generates  counter  e.m.fs.  which  are  somewhat  greater  near 
the  center  of  the  wire  than  at  the  circumference,  the  result  being 
that  the  current  density  becomes  greater  near  the  surface  of  the 
wire  than  in  the  center  portions.  This  phenomenon  is  known  as 
the  skin  effect.  The  additional  resistance  offered  to  the  passage 
of  alternating  currents,  and  the  correspondingly  increased  PR 
losses  are,  however,  small  and  generally  of  negligible  amount  on 
low  frequencies,  unless  the  cross-section  of  the  conductor  is  very 
large ;  it  is  with  the  higher  frequencies  that  this  effect  becomes  of 
importance.  Means  of  calculating  increase  of  resistance  due  to 
skin  effect  will  be  given  in  Chapter  IV. 

4.  Transmission  by  Two-phase  Currents. — If  four  separate 
wires  are  run  from  generating  to  receiving  station,  as  indicated 


R      

• I 

#2  $Load 


un&y  r  « 


H 


FIG.  2. — Two-phase  transmission  with  four  wires. 

in  Fig.  2,  and  if  the  load  is  the  same  on  both  circuits,  the  total 
power  transmitted,  on  the  assumption  of  negligible  inductance,  is 

W  =  2EI 

where  E  stands  for  the  terminal  voltage  EI  or  Ez  of  either  section 
at  the  receiving  end;  these  pressures  being  assumed  equal.  The 
pressure  lost  in  transmission  is  2R  X  /,  and  the  watts  lost  are 
2(2R  X  /2);  the  system  being  simply  a  transmission  of  energy 
by  means  of  two  independent  single-phase  circuits.  It  will  be 
seen,  however,  that,  by  combining  two  of  the  conductors  to  form 
a  common  return  path  for  the  current,  the  transmission  of  two- 
phase  currents  can  be  effected  with  only  three  wires,  as  indicated 


16  ELECTRIC  POWER  TRANSMISSION 

in  Fig.  3.  The  vector  diagram  for  such  a  system  of  transmission 
is  easy  to  construct  if  il  is  permissible  to  assume  the  resistance  R' 
of  the  common  conductor  to  be  negligible;  the  current  relations,  with 
a  quarter  period  time  displacement  between  the  currents  in  the 
two  phases,  being  as  indicated  in  Fig.  4.  The  current  Is  in  the 


R 


ri+/a    ^  I 


./,  R    i 


Fro.  3. — Two-phase  transmission  with  three  wires. 

common  conductor  is  the  vectorial  sum  of  the  currents  I\  and  72. 
It  has  been  drawn  equal  but  opposite  to  the  dotted  resultant  OA 
because  it  is  generally  convenient  to  assume  the  direction  of  all 
currents  to  be  positive  when  flowing  away  from  the  source  of 
supply,  in  which  case  the  condition 

/!   +  /2   +  /3    =    0 

A 


FIG.   4. — Vector   diagram   of   currents   in   two-phase   three-wire   transmission. 

must  be  satisfied.  The  arrow  on  the  central  conductor  in  Fig.  3 
indicates  a  flow  of  current  opposite  to  the  currents  in  the  other 
two  wires;  but  this  is  done  merely  to  suggest  the  idea  of  currents 
going  away  from  the  source  of  supply  by  the  two  outer  wires, 
and  returning  by  the  common  wire.  When  dealing  with  poly- 


ELECTRIC  PRINCIPLES  AND  THEORY  17 

phase  currents,  such  mental  pictures  of  the  actual  physical  oc- 
currences are  liable  to  be  misleading,  and  they  should  be  used 
sparingly. 

On  a  long  transmission,  or  in  cases  where  the  resistance  of  the 
common  conductor  cannot  be  neglected,  the  problem  of  two- 
phase  transmission  by  means  of  only  three  wires  becomes  com- 
plicated. The  resistance  of  the  common  wire  has  the  effect  of 
disturbing  the  phase  relations,  which  are  no  longer  the  same  at 
the  receiving  end  as  at  the  generating  end  of  the  line.  It  is 
difficult  to  explain  in  simple  terms  exactly  how  this  occurs; 
and  since  very  few  systems  of  transmission  are  by  means  of  two- 
phase  currents  it  has  not  been  thought  necessary  to  take  up 
space  in  the  study  of  this  peculiarity.  Those  interested  in  the 
question  will  find  a  complete  discussion  of  the  problem  of  two- 
phase  transmission  by  means  of  three  conductors  in  Appendix  C 
of  the  first  edition  of  this  book.  On  the  few  long-distance 
transmissions  by  two-phase  currents,  that  are  operating  suc- 
cessfully, four  conductors  are  used.  The  advantage  of  three 
wires  over  four  is,  of  course,'  in  the  saving  of  cost  on  the 
conductors.  Instead  of  requiring  four  wires,  each  carrying  I 
amperes,  it  is  only  necessary  to  provide  two  wires  to  carry  7 
amperes  and  one  wire  of  a  cross-section  sufficient  to  carry, 
not  27  amperes,  but  \/72  -{-  72  or  7  X  \/2  amperes;  which  leads 
to  an  appreciable  saving  in  the  total  weight  of  conductors,  if  the 
current  density  is  the  same  in  all. 

5.  Transmission  by  Three-phase  Currents. — If  six  separate 
conductors  are  run  from  generating  to  receiving  station,  as 
indicated  in  Fig.  5,  the  transmission  is  equivalent  to  three  in- 
dependent single-phase  two- wire  circuits;  and  if  En  is  the  poten- 
tial difference  at  the  terminals  of  each  circuit,  and  7  the  current 
in  each  wire,  the  total  power  transmitted  will  be 

W  =  Z(En  X  7) 

the  assumption  being,  as  in  previous  cases,  that  both  inductance 
and  capacity  are  of  negligible  amount. 

The  pressure  lost  in  transmission  will  be  2R  X  7  and  the  total 
power  lost  in  the  three  lines  will  be  3  X  72  X  2R. 

Consider  now  the  arrangement  as  in  Fig.  6,  where  the  three 
circuits  have  a  common  terminal  at  each  end  of  the  transmission 
and  three  of  the  wires  of  the  six-wire  transmission  are  replaced 
by  a  common  return  conductor.  The  pressure  at  the  receiving 

2 


18 


ELECTRIC  POWER  TRANSMISSION 


end,  between  each  of  the  three  terminals  and  the  common  return, 
or  neutral  point,  is  still  En  volts;  and  the  total  power  transmitted 
is  still  W  =  3(En  X  /);  but,  owing  to  the  fact  that  the  sum  of 
the  three  outgoing  currents  is  zero  (since  they  differ  in  phase  by 
120  time-degrees,  as  shown  in  Fig.  7,  and  any  one  current,  such 
as  OB,  is  exactly  equal  and  opposite  to  the  resultant  of  the  other 


Fio.  5. — Three-phase  transmission  with  six  wires. 

two  currents),  there  will  be  no  current  flowing  in  the  common 
return  conductor,  which  can  therefore  be  omitted;  and  it  follows 
that  both  pressure  drop  and  PR  losses  in  the  lines  are  reduced 
to  one-half  of  what  they  were  with  the  arrangement  of  three 
separate  circuits;  the  power  loss  in  the  lines  being  now  3PR. 
This  clearly  shows  how  the  transmission  by  three-phase  currents 
is  more  economical  as  regards  line  losses  than  single-phase  trans- 


Fia.  6. — Three-phase  transmission  with  three  wires. 

mission.  But  it  must  not  be  overlooked  that,  in  order  to  obtain 
a  reduction  by  half  of  the  weight  of  copper  in  the  lines,  the  pres- 
sure between  the  wires  is  greater  on  a  three-phase  system  than 
on  a  single-phase  system  transmitting  the  same  amount  of  power. 
Thus,  the  pressure,  V  (Fig.  6),  between  any  two  of  the  three 
transmission  wires  is  the  difference  between  two  of  the  star  volt- 
ages, as  indicated  in  Fig.  8.  Here  the  e.m.fs.  in  the  three  sections 


ELECTRIC  PRINCIPLES  AND  THEORY 


19 


of  the  alternator  windings  are  represented  by  the  vectors  OA, 
OB,  and  OC;  and  since  the  e.m.f.,  V,  between  any  two  terminals, 
such  as  B  and  C  (Fig.  6),  is  the  resultant  of  the  e.m.f s.  acting 
in  the  two  windings  OB  and  OC  connected  in  series,  one  of  these 
(as  OC)  must  be  subtracted  from  the  other  (OB).  Thus  the 
resultant  is  the  vector  0V  (Fig.  8),  obtained  by  adding  to  the 
vector  OB  an  imaginary  vector,  OC',  exactly  equal,  but  opposite, 
to  OC.  This  resultant  is  evidently  equal  and  parallel  to  the  line 
CB,  joining  the  ends  of  the  two  vectors  OB  and  OC,  and  since  the 

A  ^       ^  C" 


FIG.  7. — Vector  diagram  of  currents 
in  three-phase  transmission. 


FIG.  8. — Vector  diagram  of  e.m.fs.  in 
three-phase  star-connected  system. 


angle  C'OV  is  30  degrees,  the  length  0V  is  equal  to  20C  cos  30°, 
or  \/3  times  the  length  of  any  one  of  the  vectors  representing 
e.m.fs.  between  conductor  and  neutral  point.  Thus, 

V  =  1.732Fn 

and  since  the  resistance  drop  is  the  same  in  all  three  conductors, 
a  similar  condition  exists  at  the  receiving  end  of  the  line,  and 
we  may  also  write, 

E  =  1.732#n  (1) 

The  power  of  a  three-phase  circuit,  which  is  three  times  En  XI, 
can  evidently  alsc  be  written 

W  =  3/4=  X 

\V'3 
or, 

w  =  VSEI  (2) 

where  E  is  the  pressure  between  any  two  of  the  three  wires.1 

6.  Relative   Cost   of   Conductors   Required   on   the   Various 

Systems. — Apart  from  all  questions  of  voltage,   or  necessary 

insulation   and   spacing   required   between  adjacent  conductors 

1  The  power  factor  (cos  6)  does  not  appear  in  this  formula  because,  owing 
to  the  assumed  absence  of  inductance  or  capacity  from  both  line  and  load, 
it  is  equal  to  unity. 


20  ELECTRIC  POWER  TRANSMISSION 

and  between  the  conductors  and  the  supporting  structures,  the 
total  PR  losses  will  be  the  sum  of  the  losses  occurring  in  each 
conductor  of  the  transmission  system.  Each  wire  may  be  con- 
sidered as  the  outgoing  conductor  of  a  two-wire  single-phase 
system  in  which  the  return  wire  has  no  resistance.  Thus,  in  a 
balanced  three-phase  system  as  illustrated  in  Fig.  6,  wherein 
the  common  return  wire  is  not  required  (since  it  carries  no  cur- 
rent), the  total  losses  in  the  transmission  wires  are 

3(/2  X  K) 
But  this  may  be  written, 

(3/)2  X  | 

which  shows  that  the  total  losses  can  be  calculated  by  adding 
the  currents  in  the  respective  conductors  regardless  of  phase 
relations,  and  considering  this  total  current  as  being  transmitted 
over  a  single  wire  of  the  same  weight  or  cross-section  as  would  be 
obtained  by  connecting  the  individual  conductors  in  parallel. 
This  applies  to  any  polyphase  system  with  wires  of  equal  resist- 
ance carrying  equal  amounts  of  current. 

When  comparing  different  systems  of  transmission,  it  is  nec- 
essary to  make  some  assumptions  in  regard  to  the  voltage, 
so  as  to  have  a  common  basis  of  comparison.  For  instance,  if 
it  is  desired  to  compare  three-phase  and  single-phase  transmission 
on  the  basis  of  the  same  potential  difference  between  wires,  apart 
from  any  question  of  voltage  between  wires  and  ground,  the  total 
power  on  the  three-phase  system  will  be 

W  =  v'ZE  X  I 


and  the   (equal)   power  on  the  single-phase  system  would  be 
written, 

W  =  E  X  (-N/3/) 
Let  Rs  =  the  resistance  of  each  conductor  on  the  three-phase 

system,  and 

Ri  =  the  resistance  of  each  conductor  on  the  single-phase 
system, 

then,  for  equal  total  line  losses, 


whence 

Ri  =  2R1 


ELECTRIC  PRINCIPLES  AND  THEORY 


21 


Since  the  weight  of  copper  in  each  wire  of  either  system  is 
inversely  proportional  to  the  resistance,  it  follows  that, 


Weight  of  copper,  single-phase 
Weight  of  copper,  three-phase 


2R3 
3R  i 
22/20 


4 
~3 

which  indicates  a  saving  of  25  per  cent,  of  conductor  material 
in  favor  of  the  three-phase  system. 

Consider  now  the  condition  of  the  various  systems  on  the 
basis  of  the  same  efficiency  (as  in  the  above  example),  but  on  the 


Ground 

V 

Single-Phase  Two  (  or  Four  )  Phase  Three-Phase 

FIG.  9. — Different  systems  with  same  potential  above  ground. 

further  assumption  that  the  potential  difference  between  the 
earth  or  supporting  structures  and  any  one  of  the  conductors  is 
constant.  This  is  equivalent  to  stating  that  the  pressure  stress 
at  every  point  of  support,  where  an  insulator  carries  the  conduc- 
tor, is  the  same  on  all  systems.  This  is  shown  diagrammatically 
in  Fig.  9,  where  the  voltage  En  per  phase  is  the  same  in  all  systems. 
If  /  is  the  current  per  wire,  and  n  the  number  of  wires  (or 
phases),  the  total  power  transmitted  is 

W  =  EJ  X  n 

provided  the  power  factor  is  unity.  On  the  assumption  of  a 
balanced  load,  with  the  current  lagging  behind  the  voltage  by 
the  same  number  of  time-degrees  on  each  phase  of  all  the  systems, 
no  complication  will  arise  if  the  power  factor  of  the  load  is  taken 


22  ELECTRIC  POWER  TRANSMISSION 

into  account.     The  total  power  transmitted,  in  every  case,  may, 
therefore,  be  written 

W  =  EJ  cos  6  X  n  (3) 

If  R  is  the  resistance  of  each  line  conductor,  the  total  line  loss 
for  any  system  will  be 

w  =  PR  X  n 

and,  for  the  same  line  efficiency,  the  weight  of  copper  per  kilo- 
watt transmitted  will  evidently  be  the  same  in  all  cases. 

This  leads  to  the  conclusion  that,  for  any  balanced  polyphase 
system,  the  power  lost  in  the  line  depends  only  upon  the  joint 
resistance  of  the  conductors,  the  power  transmitted,  and  the 
power  factor,  provided  the  pressure  between  the  conductors  and  the 
neutral  point  is  constant. 

Still  neglecting  the  inductance  and  capacity  of  the  line  itself, 
the  percentage  power  lost  in  transmission  is 


If  .the  loss  w  be  expressed  in  terms  of  the  total  power  W,  it 
will  be  found  that  this  ratio  can  be  put  in  an  interesting  form. 
The  symbol  Rp  will  be  used  to  denote  the  joint  resistance  of  all 
the  conductors  in  parallel;  that  is  to  say, 

7?    -R 

Rp~n 

The  power  lost  is 

w  =  nPR 

=  n*PRp  (4) 

but  for  n2  may  be  substituted  its  equivalent  value 

W2 


ey 

obtained  from  equation  (3),  whence  (4)  becomes, 

WZRP 

ni\      —  -      _  •**  _____ 

#n2(cos  0)2 

which  shows  how,  for  any  given  amount  of  power  transmitted 
at  a  given  pressure,  the  IZR  loss  is  directly  proportional  to  the 
joint  resistance  of  all  the  conductors,  and  inversely  proportional 
to  the  square  of  the  power  factor  of  the  load. 

By  substituting  this  value  for  w  in  the  ratio  for  percentage 
efficiency,  the  latter  quantity  becomes, 

Percentage  power  lost  in  any  1    _      WRP          ^         ^ 
balanced  polyphase  system  /        En2  cos2  6 


ELECTRIC  PRINCIPLES  AND  THEORY  23 

These  formulas  show  very  clearly  the  advantages  of  high  power 
factors  where  economy  of  transmission  is  important. 

7.  Grounding  the  Neutral  on  High-tension  Overhead  Trans- 
missions.— The  above  comparisons  of  losses  have  been  made  on 
the  assumption  of  a  grounded  neutral,  but  whether  this  common 
terminal  of  the  polyphase  circuit  is  grounded  or  not  does  not 
alter  the  fact  that,  under  normal  conditions  of  working,  the  neutral 
point  is  usually  at  about  the  same  potential  as  the  ground.  Fur- 
ther, whether  the  generator  or  transformer  windings  on  the  high- 
tension  side  of  a  three-phase  transmission  system  are  delta 
connected  or  star  connected  is  of  very  little  importance  if  it  is 
decided  to  operate  without  a  grounded  point.  The  fact  that  the 
neutral  point  on  a  star-connected  system  is  available  for  ground- 
ing purposes  does  not  mean  that  it  must  necessarily  be  grounded. 

The  chief  arguments  in  favor  of  grounding  the  neutral  are 
(1)  that  the  difference  of  potential  between  any  conductor  and 
the  supporting  structure  or  earth  remains  unaltered,  and  can- 
not become  excessive  in  the  event  of  the  grounding  of  a  high-ten-, 
sion  conductor,  and  (2)  that  it  is  possible  to  detect  instantly, 
and  disconnect  by  automatic  devices  or  otherwise,  any  portion 
of  the  system  that  may  become  accidentally  grounded.  The 
chief  objection  is  that  under  such  conditions,  the  grounding  of 
any  one  conductor  causes  a  short-circuit,  and  even  if  discon- 
nected by  the  opening  of  a  switch,  leads  to  an  interruption  of 
supply.  By  inserting  a  resistance  between  the  neutral  and  the 
ground  connection,  the  current  through  the  fault  can  be  limited 
to  just  so  large  an  amount  as  may  be  necessary  to  operate  an 
automatic  device,  or  give  an  indication  that  there  is  a  fault  on 
the  line.  Instead  of  opening  the  switches  and  disconnecting 
the  line,  the  ground  connection  to  neutral  may  be  opened,  thus 
leaving  the  conductor  grounded  until  such  time  as  it  is  convenient 
to  carry  out  repairs;  but  this  would  be  equivalent  to  running 
normally  without  grounded  neutral.  The  chief  advantage  of 
transmitting  with  ungrounded  neutral  is  that  the  grounding  of 
one  conductor  only  does  not  lead  to  immediate  interruption  of 
service.  The  chief  disadvantage  is  probably  the  fact  that  the 
potential  between  earth  and  the  other  conductors  is  immedi- 
ately raised;  being  \/3  times  greater,  in  the  case  of  a  three- 
phase  transmission,  than  under  normal  conditions,  when  the 
voltages  are  balanced. 

It  is  doubtful  whether  the  question  of  grounding  the  neutral 


24  ELECTRIC  POWER  TRANSMISSION 

on  a  high-tension  transmission  can  be  so  settled  as  to  be  appli- 
cable to  all  systems  and  voltages.  A  few  years  ago  there  was  an 
undoubted  tendency  on  the  part  of  engineers  to  ground  all  metal 
that  would  under  normal  conditions  be  at  ground  potential  At 
the  present  time  the  tendency  appears  to  be  in  the  direction  of 
providing  substantial  insulation  throughout  the  system,  and 
omitting  the  grounding  of  the  neutral.  In  an  age  when  the  indi- 
vidual appears  to  distrust  the  conclusions  of  his  own  intellect, 
there  would  appear  to  be  much  wisdom  in  the  advice  once  given 
by  Dr.  Steinmetz,  who  suggested  that  if  the  engineer  is  in  doubt 
as  to  the  better  course  to  pursue  when  two  alternatives  present 
themselves,  he  should  not  follow  the  one  most  favored  by  his 
fellow  engineers,  because  in  so  doing  he  would  in  all  proba- 
bility merely  adopt  what  happens  to  be  the  fad  of  the  day.  It  is 
perhaps  best  practice  to  avoid  grounding  any  point  on  a  high-ten- 
sion transmission  unless  the  conditions  are  such  that  the  ground- 
ing of  the  neutral  point  would  appear  to  be  the  obvious  remedy  for 
troubles  that  may  have  been  experienced  or  that  are  liable  to 
occur. 

Regulation — Effect  of  Line  Inductance  on  the  Transmission 
of  Alternating  Currents. — On  account  of  the  necessary  space 
between  the  wires,  the  loops  formed  between  outgoing  and  return 
conductors  are  of  considerable  area  on  a  long-distance  trans- 
mission; and  the  changing  flux  of  induction  in  these  loops  will 
generate  counter  e.m.f.'s  in  the  conductors,  which  may  be  of 
considerable  importance,  especially  in  regard  to  their  effect  on 
the  voltage  regulation.  Whether  dealing  with  single-phase  or 
polyphase  transmissions,  it  will  be  found  convenient  to  make 
calculations  on  single  conductors  only.  Thus,  instead  of  con- 
sidering the  resistance  of  the  complete  circuit  (which  is  not  con- 
venient in  the  case  of  polyphase  transmissions),  the  resistance  of 
one  conductor  only,  or  the  resistance  per  mile  of  single  conductor 
is  considered,  and  the  ohmic  voltage  drop  calculated  for  that 
portion  of  the  complete  circuit  only.  Similarly,  in  the  matter  of 
the  counter  e.m.f.  due  to  the  self-induction  of  the  line,  calcula- 
tions are  based,  not  on  the  total  flux  of  induction  in  the  loop  or 
loops  formed  by  outward-going  and  return  wires,  but  on  that 
portion  of  the  total  flux  which  is  included  between  the  center  line 
of  any  one  conductor  and  the  neutral  plane  or  line.  Thus  the 
induced  volts  per  single  conductor,  or  per  mile  of  single  conductor, 
may  be  calculated,  and  the  resulting  total  voltage  drop  may  be 


ELECTRIC  PRINCIPLES  AND  THEORY  25 

computed  for  each  conductor  independently  of  the  others.  In 
the  case  of  a  single-phase  two-wire  transmission,  the  total 
loss  of  pressure  is  evidently  just  twice  the  amount  so  arrived  at 
for  a  single  conductor.  In  a  polyphase  transmission,  due  atten- 
tion has  to  be  paid  to  the  phase  relations  between  the  currents 
in  the  various  conductors;  but  the  same  principle  holds  good, 
and  calculations  of  any  polyphase  transmission  can  be  made  by 
considering  each  conductor  separately,  as  will  be  explained  later. 
The  induced  volts  will  be  directly  proportional  to  the  current, 
and  will  depend  on  the  diameter  of  the  wire  and  its  distance  from 
the  return  conductors.  This  will  be  again  referred  to  in  Chapter 
IV,  but  for  the  present  the  induced  pressure  may  be  calculated  by 
means  of  the  following  formula : 


Volts  induced  per  mile 
of  single  conductor 


-  0.00466  X  /  X  7  X  Iog1(/1.285  ^ 


(6) 


where  d  and  r  stand  respectively  for  the  distance  between 
outward  and  return  (parallel)  conductors  and  the  radius  or  half 
diameter  of  the  wire;  these  being  expressed  in  the  same  units. 
The  frequency  /  is  expressed  in  cycles  per  second,  and  the  current 
/,  in  amperes.  In  nearly  all  pocket  books  or  hand  books  for 
the  use  of  electrical  engineers,  tables  are  published  giving  in- 
ductive pressure  drop  for  different  diameters  and  spacings  of 
wires;  the  assumption  being  always,  as  in  the  case  of  formula  (6), 
that  the  current  variation  is  in  accordance  with  the  simple 
harmonic  law  (sine  wave).  The  special  case  of  magnetic 
conductors  such  as  iron  or  steel  will  be  referred  to  in  Chapter  IV. 
9.  Fundamental  Vector  Diagram  for  Line  Calculations: 
Capacity  Neglected. — In  the  diagram  Fig.  10,  the  various  quan- 
tities are  represented  as  follows : 

OA,  or  (7),  is  the  current  vector. 

OB,  or  (En\  is  the  vector  corresponding  to  the  pressure 
(wire  to  neutral)  at  the  receiving  end. 

0  is  the  time  angle  by  which  the  current  lags  behind  the 
pressure  at  receiving  end :  cos  8  being  the  power  factor 
of  the  load. 

BC,  or  (IR),  which  is  drawn  parallel  to  OA,- is  the  quan- 
tity I  X  R;  being  the  voltage  component  required  at 
the  generating  end  to  compensate  for  ohmic  drop  of 
pressure  in  the  conductor. 


26  ELECTRIC  POWER  TRANSMISSION 

CD,  or  (IX),  which  is  drawn  at  right  angles  to  OA,  is  the 
quantity  calculated  by  formula  (6),  being  the  voltage 
component  required  at  generating  end  to  compensate 
for  loss  of  pressure  due  to  the  inductive  reactance  of  the 
conductor. 

BD  is  the  sum  of  the  vectors  BC  and  CD;  being  the  total 
additional  voltage  required  at  the  generating  end  to 
compensate  for  the  impedance  of  the  conductor. 

OD,  or  (Fn),  is  the  vector  corresponding  to  the  pressure 
(wire  to  neutral)  at  generator  end  of  the  line,  required 
to  maintain  the  pressure  (En)  at  the  receiving  end  when 
the  current  in  the  conductor  is  /  amperes. 


\D 


(/) 

FIG.  10.  —  Vector  diagram  for  line  calculations  —  capacity  neglected. 

<p  is  the  time  angle  by  which  the  current  lags  behind  the 
pressure  at  the  generating  end;  cos  <p  being  the  power 
factor  of  the  total  load  as  measured  at  generating 
end. 

FD  is  the  (arithmetical)  difference  between  pressures  at 
receiving  and  generating  ends  of  the  conductor. 
The  percentage  loss  of  pressure  being 

100  x  lengthF£) 

*  length  OF 


100  X 
* 


length  OB 
(The  dotted  circles  being  described  from  the  center  0.) 


ELECTRIC  PRINCIPLES  AND  THEORY  27 

All  graphical  solutions  of  transmission  line  problems  are  based 
on  this  fundamental  diagram.  Some  of  them  give  results  that 
are  theoretically  correct,  while  in  others  certain  assumptions 
are  made  to  simplify  the  construction  without  introducing  any 
appreciable  error  in  the  solution  of  practical  problems. 

Graphical  and  semi-graphical  methods  of  predetermining  the 
voltage  regulation  of  transmission  lines  are  often  convenient. 
Such  methods  have  been  proposed  by  Messrs.  F.  A.  C.  Perrine 
and  F.  G.  Baum,  by  Prof.  L.  A.  Herdt,  by  Mr.  R.  D.  Mershon 
and  others.  A  convenient  diagram  for  determining  regulation, 
as  used  by  the  writer,  will  be  explained  in  Chapter  IV. 

For  those  who  prefer  to  use  tables  of  trigonometrical  functions, 
the  required  relations  can  easily  be  obtained  from  Fig.  10. 

In  the  first  place,  the  functions  of  the  angle  <p  are: 

IX  +  En  sin  6 
sin  <p  =  -    -^-  (7) 

IR  +  En  cos  6 
cos  <p  =  -    — ^—  (8) 

IX  +  En  sin  0 
tan  *  =  IR  +  En  cos  0 

From  formula  (8)  it  is  seen  that  the  required  voltage  at  generat- 
ing end  is, 

'"'^Sr1-'  <«» 

and  the  volts  required  to  overcome  ohmic  resistance  are: 

IR  =  Vn  cos  <p  -  En  cos  0  (11) 

As  an  example  of  the  use  of  these  formulas  assume,  in  the 
first  place,  that  the  material,  size  and  spacing  of  conductors  is 
known,  and  also — in  all  cases — the  power  factor  of  the  load 
(cos  0),  and  therefore  the  other  trigonometrical  functions  of 
the  angle  0,  such  as  sin  $.  Under  these  conditions,  the  quan- 
tities IR  and  IX  can  readily  be  calculated,  and  formula  (9) 
can  be  used  to  obtain  tan  <p;  thence  the  angle  <p  and  cos  <p  (the 
power  factor  at  generating  end).  Then,  by  formula  (10),  the 
required  voltage  (Vn)  at  generating  end  is  easily  obtained. 

Assume,  in  the  second  place,  that  the  size  of  the  conduc- 
tors has  to  be  determined.  The  spacing  of  conductors  and  the 
frequency  being  known,  the  induced  volts  IX  can  be  calculated 
approximately  by  estimating  the  value  of  r  for  use  in  formula 


28  ELECTRIC  POWER  TRANSMISSION 

(6).  The  more  correct  estimate  of  the  size  of  conductor  will  he 
based  on  the  required  regulation,  or  total  voltage  drop.  The 
voltage  at  the  generating  end  is  therefore 

Vn  =  En  +  allowable  voltage  drop  per  conductor 

Now,  since  IR  is  not  definitely  known,  formula  (7)  will  have  to 
be  used.  This  gives  the  value  of  sin  <p  with  a  sufficient  degree  of 
accuracy  even  if  quite  an  appreciable  error  has  been  made  in 
estimating  the  size  of  conductor  for  the  purpose  of  calculating 
the  inductive  drop  IX.  Having  determined  the  angle  <p,  the 
function  cos  <p  can  be  obtained  from  trigonometrical  tables;  and 
then,  by  using  formula  (11),  the  ohmic  drop  can  be  calculated. 
Thus  the  proper  size  of  wire  for  use  under  given  conditions  may 
be  determined. 

If  the  power  loss  in  the  line  is  the  determining  quantity,  regard- 
less of  the  voltage  regulation,  then,  since  this  loss  depends  only 
on  the  voltage  IR  (the  current,  7,  being  assumed  constant),  the 
resistance  and  size  of  conductor  is  readily  ascertained,  and  the 
unknown  quantities  would  be  calculated  as  in  the  case  first 
considered. 

10.  Effect  of  Capacity  on  Regulation  and  Line  Losses. — 
Although  the  effects  of  electrostatic  capacity  will  be  referred  to 
again  in  Chapter  IV,  it  will  be  well  to  consider  briefly  how  the 
capacity  on  long  lines  may  affect  the  voltage  regulation  and  line 
losses. 

Any  arrangement  of  two  conductors  of  electricity  separated 
by  an  insulator,  forms  a  condenser,  of  which  the  capacity  will 
depend  upon  the  spacing  of  the  conductors,  and  the  nature  of 
the  dielectric  between  them.  In  the  case  of  overhead  conductors 
running  parallel  to  each  other  and  to  the  surface  of  the  ground 
over  a  considerable  distance,  the  electrostatic  capacity  between 
the  individual  conductors,  and  between  these  conductors  and 
earth,  becomes  a  matter  of  importance. 

As  in  the  case  of  inductance  calculations,  it  is  advisable, 
whenever  possible,  to  consider  the  capacity  of  any  one  conductor 
as  measured  between  the  conductor  and  the  neutral  surface  or 
neutral  line.  Thus  the  capacity  current  per  conductor  can  be 
calculated  independently  of  the  current  in  the  other  conductors. 
It  is  obvious  that  the  potential  difference  causing  the  flow  of  cur- 
rent in  and  out  of  the  condenser  must  then  be  measured  between 
the  conductor  and  the  neutral,  and  not  between  outgoing  and 


ELECTRIC  PRINCIPLES  AND  THEORY  29 

return  conductors.  In  the  case  of  a  transmission  line,  the 
capacity  is  distributed  over  the  whole  length  of  the  line.  It  is 
incorrect  to  assume  that  the  whole  of  this  capacity  is  concen- 
trated at  either  end;  but,  for  the  sake  of  simplicity,  the  total 
capacity  will  be  supposed  to  be  concentrated  at  the  receiving 
end  of  the  line,  and  a  correction  will  afterward  be  made  in  order 
to  conform  more  nearly  to  actual  conditions. 

The  following  approximate  formula  may  be  used  for  calculat- 
ing the  capacity  of  overhead  lines: 

Capacity  in  microfarads  per  |  ft  ftQ«s 

mile,  between  conductors  }  =  Cm  =  j  (12) 

and  neutral 

where  d  and  r  are  the  spacing  between  conductors  and  the  radius 
of  cross-section,  exactly  as  in  formula  (6)  for  the  calculation  of 
the  induced  volts. 
A 


O      ~L  (En)  B 

FIG.  11. — Vector  diagram  showing  pressure  rise  due  to  capacity. 

The  charging  current,  in  amperes,  on  the  sine-wave  assump- 
tion, can  be  calculated  by  the  formula 

Jc  =  2irfCmLEn  X  10~6  (13) 

where  L  is  the  distance  of  transmission  in  miles,  and  En  is  the 
voltage  as  measured  between  the  conductor  and  neutral.  This 
charging  current  (7C)  is  always  a  quarter  period  in  advance  of  the 
voltage.  This  explains  why,  on  a  long  line  open  at  the  distant 
end,  or  only  very  lightly  loaded,  there  can  be  a  rise  of  pressure 
at  the  receiving  end  of  the  line. 

In  the  diagram  Fig.  11,  the  pressure  at  the  receiving  end  is 
represented  by  the  vector  OB,  while  OA,  drawn  at  right  angles 
to  OB — in  the  forward  direction — is  the  capacity  current  as  cal- 


30  ELECTRIC  POWER  TRANSMISSION 

culated  by  formula  (13).  It  is  assumed  that  the  load  is  entirely 
disconnected,  and  the  current  Ic  is  the  total  current  on  the  lino. 
The  voltage  component  required  at  generating  end  to  overcome 
ohmic  resistance  is  OR,  or  BC,  in  phase  with  Ic,  and  the  com- 
ponent required  to  balance  the  e.m.f.  of  self-induction,  as  cal- 
culated by  formula  (6),  is  CD,  drawn  90  degrees  in  advance  of 
OA.  The  pressure  required  at  generating  end  is  OD,  which  may 
be  smaller  than  OB.  It  is  true  that  the  capacity  has  been  as- 
sumed to  be  concentrated  at  the  receiving  end  of  the  line;  but 
with  distributed  capacity,  the  same  effect  of  a  rise  in  pressure  as 
the  distance  from  generating  end  increases,  will  occur.  It  will 
be  seen  that  this  is  due  to  the  e.m.f.  of  self-induction  of  the 
charging  current  being  in  phase  with  the  impressed  voltage.  If 
the  lines  were  without  inductance,  there  could  be  no  pressure 
rise. 

The  effect  of  capacity  on  the  line  when  the  distant  end  is  closed 
on  the  load,  will  depend  upon  the  amount  and  nature  of  the 
load.  If  the  load  is  heavy  and  largely  inductive,  the  current 
put  into  the  line  at  the  generating  end  will  be  less  than  the  load 
current,  and  the  PR  losses  will  therefore  be  smaller  than  if  the 
line  were  without  capacity. 

At  light  loads,  especially  if  the  power  factor  is  high,  the  line 
losses  will  be  greater  than  if  capacity  were  not  present.  On  many 
overhead  lines  the  effects  of  capacity  are  almost  negligible;  but  on 
long  high-voltage  lines,  these  effects  become  of  great  importance. 
As  an  example  of  capacity  effects  on  long  transmission  lines, 
consider  the  system  of  the  Southern  Power  Company  which 
transmits  energy  at  100,000  volts  (three-phase)  over  a  distance 
of  210  miles,  from  Great  Falls,  S.  C.  to  Durham,  N.  C.  Both 
copper  and  aluminum  conductors  are  used,  the  average  diameter 
of  which  is  about  0.4  in.  while  the  spacing  between  wires  is  124 
in.;  the  frequency  being  60.  The  above  data  and  formulas  (12) 
and  (13)  will  enable  us  to  calculate  the  capacity  current.  The 
numerical  values  for  use  in  the  calculation  are, 

d  =  124  in. 
r  =  0.2  in. 
/  =  60 
L  =  210 

100,000 


ELECTRIC  PRINCIPLES  AND  THEORY  31 

The  capacity  in  microfarads  per  mile  (between  conductor  and 
neutral)  by  formula  (12)  is, 

0.0388 


0.0139 


i 

logic 


The  charging  current,  by  formula  (13)  is, 

j   =  27r  X  60  X  0.0139  X  210  X  100,000 

1,000,000  X  V3 
=  63.6  amperes. 

It  follows  that  the  kilovolt-amperes    (or  apparent  kilowatts) 
put  into  the  line  at  the  generating  station  end,  when  all  the 
switches  at  the  receiving  end  are  open,  will  be  of  the  order  of 
\/3  X  100,000  X  63.6  _ 

1000 

.  The  effects   of  capacity  under  various   conditions  are   best 
studied  by  constructing  vector  diagrams. 


FIG.  12. — Transmission  line  with  concentrated  capacity. 

In  Fig.  12,  the  current,  I,  is  delivered  to  the  line  at  the  pressure 
V:  each  conductor  has  both  resistance  and  inductive  reactance, 
giving  a  pressure  E  at  the  distant  end,  where  the  whole  of  the 
capacity,  C,  is  supposed  to  be  shunted  across  the  wires.  The 
load  current  is  7. 

In  the  vector  diagram,  Fig.  13  (which  may  with  advantage  be 
compared  with  Fig.  10),  OB  and  OA  represent  respectively  the 
potential  difference  and  current  at  the  receiving  end.  The 
impressed  voltage  at  the  terminals  of  the  imaginary  condenser 
C  (Fig.  12),  will  therefore  be  E  volts,  and  the  vector  for  the  con- 
denser current  must  be  drawn  90  degrees  in  advance  of  OB; 
this  is  the  vector  ON.  The  total  current  put  into  the  line  at  the 
generating  end  will  be  OM,  which  is  the  (vectorial)  sum  of  the 
currents  I  and  Jc.  The  pressure  at  generating  end  is  made  up 
of  three  components; 


32  ELECTRIC  POWER  TRANSMISSION 

OB,  the  pressure  available  at  receiving  end; 

BC,  the  pressure  required  to  overcome  resistance  (drawn 
parallel  to  the  total-current  vector  OM) ; 

CD,  the  pressure  required  to  counteract  the  e.m.f.  of  self- 
induction  (drawn  at  right  angles  to  OM}. 

By  varying  the  angle  6  and  the  length  of  the  current  vector 
OA,  the  effect  of  the  capacity  current  with  different  power  fac- 
tors and  loads  can  easily  be  studied. 

This  method  of  correcting  the  fundamental  diagram  to  take 
account  of  capacity,  is  not  theoretically  accurate,  because  the 


O  (I) 

FIG.  13. — Vector  diagram  for  transmission  line  of  appreciable  capacity. 

capacity  is  never  concentrated  at  one  point  of  the  line;  but 
distributed  over  the  whole  distance  of  transmission. 

11.  Use  of  Fundamental  Diagram  for  Three-phase  Calcu- 
lations.— The  vector  diagram,  Fig.  14,  shows  the  relative  phases 
of  current  and  e.m.f.  for  a  three-phase  system  with  balanced  load 
when  the  power  factor  is  unity.  Here  the  three  current  vectors 
are  OA,  OB,  and  OC.  The  "star"  voltages  are, 

Oa  =  Ob  =  Oc  =  En 

each  being  in  phase  with  the  corresponding  line  current;  and 
the  voltages  measured  between  the  three  conductors  of  the  trans- 
mission line  are, 

ab  =  be  =  ca  =  E  =  \/3En 


ELECTRIC  PRINCIPLES  AND  THEORY 


33 


The  total  power  transmitted  is, 


W  =       3E  X  I 
=  ZEn  X  I 
=  3  (OA  X  Oa) 

In  Fig.  15  the  diagram  has  been  drawn  for  an  inductive  load. 
Here  there  is  a  certain  displacement  of  the  current  phases  rela- 
tively to  the  e.m.f.  phases.  It  will  be  noticed  that  the  ver- 
tices of  the  e.m.f.  triangle  no  longer  lie  on  the  current  lines  as  in 
the  previous  diagram.  The  three  current  vectors  still  subtend 
the  same  angle  of  120  degrees  with  each  other;  but  they  have 
been  moved  bodily  round  (in  the  direction  of  retardation)  through 


i 


FIG.  14. — Vector  diagram  for  FIG.  15. — Vector  diagram  for  three-phase 
three-phase  system  on  non-induc-  system  on  partly  inductive  load, 

tive  load. 

an  angle  6.  The  total  power  is  evidently  no  longer  equal  to 
three  times  OA  X  Oa,  but  to  3  X  OA'  X  Oa,  where  OA'  is  the 
projection  of  OA  on  Oa;  and  cos  6  is  the  power  factor  of  the  three-' 
phase  load. 

It  is  a  simple  matter  to  complete  this  diagram  by  taking  into 
account  the  effects  of  resistance  and  inductance  in  the  line, 
because  when  the  calculations  for  resistance  drop  and  induced 
volts  are  made  per  conductor  as  previously  explained,  the  construc- 
tion can  be  carried  out  for  each  phase  exactly  as  explained  when 
describing  the  fundamental  diagram  (Article  9).  It  is  only 
necessary  to  bear  in  mind  that  OA  and  Oa,  in  Fig.  15,  cor- 
respond to  OA  and  OB  in  Fig.  10.  When  this  construction 


34 


ELECTRIC  POWER  TRANSMISSION 


has  been  carried  out  for  each  of  the  three  phases,  there  will  be  a 
new  set  of  star  vectors  which,  when  their  ends  are  joined,  will  form 
a  new  e.m.f.  triangle  representing  the  necessary  pressures  at  the 
generating  end.  This  is  shown  in  Fig.  16,  where  am  and  md  are 
the  vectors  representing  the  required  e.m.f.  components  to  coun- 
teract the  ohmic  drop  and  reactive  voltage  respectively,  due  to 
the  current  OA.  The  same  construction  is  supposed  to  be  fol- 
lowed for  the  other  two  phases,  and  the  resulting  triangle  def 
indicates  not  only  the  magnitude  of  the  potential  differences  be- 
tween wires  at  generating  end,  but  also  their  phase  relations  with 
the  other  quantities.  Thus  the  power  factor  at  the  generating 


FIG.  16. — Complete  vector  diagram  for  three-phase  transmission. 

end  is  not  cos  0,  but  cos  <p,  all  as  explained  in  connection  with 
Fig.  10.  If  it  is  required  to  take  into  account  the  effects  of  ca- 
pacity, the  correction  per  phase  is  made  as  explained  in  Article  10. 
It  is  true  that  a  symmetrical  arrangement  of  conductors  has 
been  assumed;  that  is  to  say,  the  three  conductors  are  supposed 
to  occupy  the  vertices  of  an  equilateral  triangle,  in  which  case 
the  magnetic  flux  due  to  the  current  in  one  of  the  wires  will 
neither  increase  nor  decrease  the  amount  of  induction  through 
the  loop  formed  by  the  other  two  wires;  or,  in  other  words,  the 
whole  of  the  current  in  any  one  conductor  may  be  considered  as 
returning  at  a  distance  from  this  conductor  equal  to  the  side  of  the 
equilateral  triangle.  As  a  matter  of  fact,  if  the  wires  are  arranged 
in  any  other  practical  manner,  the  effect  of  the  induction  due  to 
any  one  wire  on  the  loop  formed  by  the  other  two  wires  is  usually 


ELECTRIC  PRINCIPLES  AND  THEORY  35 

small;  but  a  method  of  calculating  the  induced  volts  in  any  one 
conductor  of  a  system  of  parallel  conductors,  whatever  may  be 
the  arrangement  or  spacing  of  these  conductors,  is  explained  in 
Appendix  I  at  the  end  of  this  book. 

If  the  conductors  of  a  three-phase  transmission  are  regularly 
transposed,  that  is  to  say,  if  each  of  the  three  wires  occupies  a 
particular  position  relatively  to  the  other  wires  for  one-third  of 
the  total  length  of  transmission,  then  the  electrical  calculations 
may  be  based  on  an  equivalent  disposition  of  the  wires  at  the  ver- 
tices of  an  equilateral  triangle  of  side 

d  =  \fabc  (14) 

where  a,  b  and  c  stand  respectively  for  the  actual  spacings  between 
the  three  wires.  This  is  a  general  statement  of  the  particular 
problem  considered  in  Appendix  I,  where  the  three  conductors 
are  assumed  to  lie  in  the  same  plane.  A  neat  proof  of  formula 
(14)  is  given  in  Prof.  H.  B.  D  wight's  book  on  Transmission  Line 
Formulas.1 

1  D.  Van  Nostrand  Company,  1913. 


CHAPTER  III 
ECONOMIC  PRINCIPLES  AND  CALCULATIONS 

12.  Introductory. — That  true  engineering  is  essentially  an 
economic  science  should  be  self-evident  to  every  man  who  lays 
claim  to  the.  title  of  engineer;  yet,  there  are  many  engineering 
undertakings,  or  portions  of  such  undertakings,  in  which  this 
fundamental  principle  has  been  disregarded.  In  the  case  of 
transmission  lines,  a  certain  system,  or  an  exceptionally  high 
pressure,  may  have  been  adopted  because  of  its  peculiar  interest 
as  an  engineering  problem;  or  duplicate  lines,  spare  generating 
plant,  and  costly  automatic  gear  may  have  been  installed  to 
ensure  continuity  of  supply,  apart  from  the  economic  value 
of  such  increased  protection  against  possible  interruption.  This, 
however,  is  not  engineering  in  the  commercial  sense.  The  de- 
termination of  the  economical  size  of  conductors  for  the  transmis- 
sion of  any  particular  amount  of  current,  in  accordance  with  the 
principle  generally  known  as  Kelvin's  law,  is  but  a  very  small 
part  of  the  problem  to  be  solved  by  the  transmission-line 
engineer.  An  attempt  will  be  made  in  this  chapter  to  deal  with 
the  economics  of  power  transmission  lines  from  a  broad  practical 
standpoint.  Some  approximate  figures  for  use  in  getting  out 
preliminary  estimates  will  be  given,  but  actual  recorded  costs 
of  finished  work  carried  out  under  various  conditions  can  be 
obtained  from  other  sources;  their  inclusion  in  this  book  might 
tend  to  confuse  rather  than  assist  the  reader. 

It  is  proposed  to  deal  here,  in  as  small  a  space  as  possible, 
with  economic  principles,  and  to  explain  the  methods  by  which 
the  proper  size  of  conductor  for  a  given  transmission  can  be  cal- 
culated; and  it  is  only  when  these  principles  have  been  grasped, 
and  rightly  understood,  that  the  engineer  can  make  the  best  use 
of  cost  data  obtained  from  completed  work. 

When  considering  any  scheme  of  power  transmission  from  a 
generating  plant  of  limited  output,  it  is  important  to  bear  in 
mind  that  it  does  not  pay  to  cover  a  distance  greater  than  that 
within  which  there  is  a  reasonable  prospect  of  supplying  all  the 

36 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    37 

power  available  at  the  generating  station.  The  importance  of 
this  principle  should  be  fairly  obvious,  yet  there  are  instances 
which  prove  that  it  has  been  disregarded. 

On  the  other  hand,  where  the  energy  available  appears  to  be 
in  excess  of  the  probable  demand  within  a  reasonable  radius 
from  the  generating  station,  the  possibilities  of  subsequent 
requirements  greatly  exceeding  the  immediate  demand  must  not 
be  overlooked;  and  a  transmission  system  on  a  large  scale, 
designed  to  satisfy  future  conditions,  may  be  desirable.  Each 
case  must  be  studied  separately,  on  account  of  the  variable 
nature  of  the  conditions  in  different  localities.  The  easiest  part 
of  the  problem  is  the  designing  of  an  economical  transmission 
line,  that  is  to  say,  a  line  that  will  give  the  best  return  for  capital 
invested,  on  the  assumptions  of  a  given  amount  of  energy  to  be 
delivered  at  a  given  point  where  a  given  price  will  be  paid  for  it. 
The  real  difficulty  lies  in  estimating  not  only  the  immediate 
demand,  but  also  the  probable  future  demand  and  its  rate  of 
growth,  in  order  that  proper  provision  may  be  made  to  avoid  un- 
necessary waste  in  remodelling  or  reconstructing  the  original 
transmission  line. 

13.  Choice  of  System. — On  this  continent  it  is  usual  to  trans- 
mit electric  power  by  means  of  three-phase  alternating  currents, 
the  periodicity  being  25  or  60  cycles  per  second.     In  Europe  the 
Thury  system  of  continuous  current  transmission  at  high  voltages 
has  met  with  success;  it  has  much  to  recommend  it,  and  there 
appear  to  be  no  reasons  why  it  should  not  meet  with  equal 
success  on  this  continent;  but  it  is  probable  that  three-phase 
transmission,  at  pressures  even  higher  than  those  now  in  use, 
will  hold  its  own  for  a  considerable  time  to  come. 

14.  Type  of  Transmission  Line. — The  structures  for  supporting 
overhead  conductors  may  be  of  wood,  steel,  or  reinforced  concrete. 
The  wood  supports  may  consist  of  single  poles  spaced  120  to  300 
ft.  apart,  or  they  may  be  A  or  H  frames  built  up  of  two  poles 
suitably  braced,  and  capable  of  supporting  longer  spans.     The 
steel  poles  may  be  of  simple  tubular  type,  or  built  up  of  several 
tubes  or  angles  with  the  necessary  bracing.     The  more  common 
construction   for    high-pressure    transmission   lines    consists   of 
light-braced  towers  with  wide  rectangular  bases,  except  where  the 
"flexible"  type  of  structure  is  adopted.     These  flexible  towers  are 
modeled  generally  on  the  A  and  H  types  of  double  wood  pole 
supports.     Further    particulars,    with    illustrations,    of    these 


38  ELECTRIC  POWER  TRANSMISSION 

different  kinds  of  supports  will  be  found  in  Chapter  X.  It  is  by 
no  means  an  easy  matter  to  decide  upon  the  most  suitable  type 
of  supporting  structure  to  be  used  on  any  particular  transmission 
scheme.  In  some  cases  a  composite  line  including  two  or  more 
types  of  support  may  be  found  advantageous.  Among  the 
factors  influencing  the  choice  of  the  supporting  structures  may  be 
mentioned  the  character  of  the  country,  the  means  and  facilities 
of  transport,  climatic  conditions,  the  nature  of  the  soil,  and  the 
scarcity  or  otherwise  of  suitable  timber  in  the  district  through 
which  the  line  will  pass.  In  undulating  or  hilly  country,  advan- 
tage may  frequently  be  taken  of  the  heights,  by  erecting  upon 
them  comparatively  low  and  cheap  structures  and  spanning  the 
depressions  or  valleys  without  any  intermediate  supports.  The 
engineering  features  must,  however,  be  very  carefully  studied  in 
all  such  exceptional  cases. 

When  transmission  lines,  or  portions  of  transmission  lines, 
have  to  be  laid  underground,  the  thickness  and  nature  of  the 
insulation,  and  the  method  of  laying,  will  affect  not  only  the  first 
cost,  but  also  the  maintenance  charges  and  the  life  of  the  cables. 
All  these  factors  must  be  taken  into  account  when  deciding  upon 
the  most  economical  type  of  construction  under  the  varying 
conditions  that  are  likely  to  occur  in  practice.  It  is  usual  to 
draw  the  insulated  conductors  into  some  form  of  duct  or  conduit; 
but  conditions  may  be  found  under  which  it  would  be  justifiable 
to  use  lead-sheathed  and  steel-armored  cable  laid  directly  in 
the  ground  without  any  additional  protection.  The  question  of 
underground  cables  will  be  discussed  further  in  Chapter  VII. 

15.  Length  of  Span. — The  type  of  supporting  structure  for 
overhead  conductors,  together  with  the  height,  strength  and  cost, 
of  the  individual  pole  or  tower,  will  be  dependent  upon  the  span 
or  distance  between  the  supports.  The  determination  of  the 
average  length  of  span  is  indeed  a  very  important  economic 
question.  The  material  of  the  conductor  will,  to  some  extent, 
influence  the  choice  of  span  length,  because  aluminum  conductors 
will  usually  have  a  greater  summer  sag  than  copper  conductors, 
and  this  will  necessitate  higher  supports  to  give  the  same  clear- 
ance above  ground  at  the  lowest  point  of  the  span.  In  consider- 
ing span  length,  the  first  cost  of  the  individual  support  is  not  the 
only  question  which  has  to  be  taken  into  account;  the  cost  of 
maintenance  is  almost  equally  important.  The  longer  the  span, 
the  fewer  will  be  the  points  of  support;  and  if  the  line  is  well 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    39 

designed  and  constructed,  there  should  be  less  trouble  through 
faults  at  insulators.  Again,  where  rent  has  to  be  paid  for  poles 
placed  on  private  property,  it  is  generally  the  rent  per  pole  apart 
from  the  size  of  pole  which  has  to  be  considered,  and  this  is 
another  factor  in  the  determination  of  the  best  length  of  span. 
In  level  country,  the  economic  span  for  steel  tower  construction  is 
usually  in  the  neighborhood  of  600  feet.  The  Sierra  &  San 
Francisco  Power  Co.  (104,000  volts)  uses  spans  of  850  feet; 
while  both  the  Pacific  Gas  &  Electric  Co.  and  Mississippi  River 
Power  Co.,  on  their  110,000-volt  transmission  lines,  space  the 
towers  800  feet  apart  under  normal  conditions.  On  short  low- 
voltage  lines  large  spans  may  not  prove  to  be  economical.  It  is 
sometimes  advantageous  to  increase  the  number  of  supports  in 
order  that  these  may  be  so  light  in  weight  as  to  be  easily  handled 
and  quickly  erected. 

When  considering  the  relative  advantages  of  different  types 
of  supports  for  overhead  conductors  it  is  obvious  that  the  life 
and  cost  of  upkeep  of  the  poles  or  towers  must  be  considered  at 
the  same  time  as  the  costs  of  delivery  on  site  and  erection, 
which  are  less  likely  to  be  overlooked.  For  instance,  a  transmis- 
sion line  using  reinforced  concrete  poles  will  usually  cost  more  than 
one  using  wood  poles;  but  if  we  assume  the  same  height  of  pole 
(say  35  feet)  and  a  cost  per  pole  of  $10  and  $6.50  respectively  for 
concrete  and  chestnut  poles,  the  concrete  pole  line  will  very 
probably  prove  to  be  the  more  economical  in  view  of  the  fact 
that  the  chestnut  poles  will  have  to  be  replaced  at  the  end  of  (say) 
13  years,  while  the  concrete  should  last,  without  requiring  atten- 
tion or  repairs,  about  twice  as  long. 

16.  Effect  of  Span  Variations  on  Cost  of  Steel  Towers. — The 
height  of  towers  in  level  country  depends  on  (1)  the  minimum 
clearance  between  the  lowest  conductor  and  ground  when  the 
sag  is  greatest;  (2)  the  voltage,  since  this  has  an  effect  on  the  spac- 
ing of  the  conductors  and  also  to  some  extent  on  the  clearance 
above  ground  level;  and  (3)  the  maximum  sag.  This  last  is 
determined  by  the  length  of  span,  the  material  and  size  of  the 
conductors,  the  range  of  temperatures,  and  weather  conditions 
generally.  For  the  purpose  of  rough  approximations  suitable 
for  preliminary  estimates  the  writer  has  made  use  of  the  em- 
pirical formula 

H  =  34  +  f  + 20^00  (15) 


40  ELECTRIC  POWER  TRANSMISSION 

This  formula  gives  the  approximate  overall  height  of  the  tower 
in  feet.  By  overall  height  is  meant  the  total  height  including 
the  portion  buried  in  the  gound;  the  height  from  ground  level  to 
extreme  top  of  tower  being  therefore  from  5  to  8  feet  less  than 
the  dimension  given  by  formula  (15).  The  symbol  Ek  stands  for 
the  working  pressure  between  wires  in  kilovolts,  while  I  is  the 
distance  between  towers  in  feet.  The  constants  have  been 
worked  out  on  the  assumption  that  the  tower  carries  a  duplicate 
three-phase  circuit  consisting  of  aluminum  conductors  of  number 
0000  B.  &  S.  gauge,  and  a  grounded  steel  guard-wire  joining  the 
tops  of  the  towers.  The  formula  is  not  supposed  to  be  appli- 
cable to  spans  greater  than  650  feet. 

The  weight  of  a  steel  supporting  structure  will  depend  not 
only  upon  the  height,  but  also  upon  the  stresses  the  tower  has  to 
withstand.  These  again  will  be  dependent  upon  the  size  of  the 
wires,  the  length  of  span,  and  the  weather  conditions,  although 
none  of  these  factors  has  perhaps  so  great  a  bearing  on  the  weight 
and  cost  of  the  supporting  structures  as  the  fancies,  prejudices, 
and  idiosyncrasies  of  the  engineer  or  purchaser.  The  assump- 
tions made  in  the  matter  of  probable  sleet  deposits  and  wind 
pressure,  together  with  the  factors  of  safety  that  are  insisted 
upon,  will  naturally  very  greatly  influence  the  cost  of  the  towers. 
Assuming  these  figures  to  be  reasonable,  the  weight  in  pounds 
of  the  complete  tower  will,  be  approximately  0.85H2,  although 
the  proportionality  between  the  square  of  the  height  and  the 
weight  does  not  always  hold  in  practice. 

The  cost  of  galvanized  towers,  under  normal  market  condi- 
tions, varies  between  3^  and  5^  per  pound.1  For  preliminary 
estimates,  the  cost  may  be  calculated  on  a  basis  of  4^  per  pound. 

The  weight  and  cost  of  towers  to  carry  only  one  three-phase 
circuit  would  be  about  25  per  cent,  less  than  double-circuit 
towers;  while  the  so-called  flexible  "A"  frame  supports  are  about 
30  per  cent,  cheaper  than  the  corresponding  rigid  square-base 
towers. 

It  is,  however,  usual  to  provide  rigid  strain  towers  in  place  of 
the  flexible  type  at,  say,  every  mile,  and  as  the  cost  of  such  struc- 

1  The  increased  price  of  all  metals,  brought  about  by  the  war,  has  led 
to  the  cost  of  steel  structures  being,  at  the  time  of  writing,  nearly  double 
what  would  be  indicated  by  these  figures,  which  are  based  on  conditions 
existing  before  1915.  It  is  not  possible  to  predict  metal  prices  for  the 
future. 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    41 

tures  is  about  double  the  cost  of  the  flexible  tower,  the  cost  of 
supports  per  mile  of  line  may  be  calculated  by  assuming  n  +  1 
flexible  structures  per  mile,  when  the  actual  spacing  is  n  to  the 
mile. 

When  estimating  the  cost  of  a  tower  line,  it  is  necessary  to 
take  into  account  the  special  anchor  towers  required  at  corners 
or  at  the  ends  of  exceptionally  long  spans.  In  any  case,  the 
practical  utility  of  the  approximate  formula  (15)  is  somewhat 
doubtful;  and  just  so  soon  as  the  investigation  has  gone  far 
enough  to  justify  the  preparation  of  a  complete  estimate  of  cost, 
quotations  for  suitable  supporting  structures  should  be  obtained 
from  the  manufacturers. 

Steel  structures  are  usually  galvanized;  but  as  an  alternative, 
they  may  be  painted;  the  extra  cost  of  galvanizing  should  be 
compared  with  the  cost  of  painting  periodically,  say,  every 
third  or  fourth  year. 

The  cost  of  foundations  for  towers  varies  greatly.  In  the 
case  of  fairly  high  steel  towers  with  wide  square  bases  in  soil 
not  requiring  the  use  of  concrete,  the  cost  of  excavating,  setting 
legs,  and  back  filling,  not  including  erection  of  towers,  will 
generally  be  between  $10.00  and  $20.00  per  tower. 

17.  Cost  of  Wood  Poles. — The  price  of  wood  poles  depends 
upon  the  kind  of  wood,  the  quality,  i.e.,  straightness  and  freedom 
from  defects,  and  of  course  upon  the  dimensions.  As  a  rough 
guide  for  preliminary  estimates,  average  prices  for  chestnut 
poles  are  given  below : 

Poles  30  ft.  long $4. 00  each 

Poles  35  ft.  long 6.00  each 

Poles  40  ft.  long 9.00  each 

Poles  45  ft.  long 10.50  each 

The  cost  of  very  tall  poles  may  be  considerable;  but  although 
the  cost  of  a  45-ft.  chestnut  pole  is  here  given  as  something 
over  $10.00,  wood  poles  of  this  length  suitable  for  transmission 
lines,  although  perhaps  less  durable,  can  be  obtained  for  $6.00 
or  even  less.  On  this  basis,  a  55-ft.  pole  would  cost  about 
$9.00,  while  70-ft.  poles  could  probably  not  be  obtained  for  less 
than  double  this  amount. 

An  amount  varying  between  $4.00  and  $6.00  should  be  allowed 
to  cover  unloading,  hauling,  "framing,"  digging  holes,  and  erect- 
ing a  pole  of  average  height. 


42 


ELECTRIC  POWER  TRANSMISSION 


Some  costs  of  sundry  items  to  be  included  in  an  estimate  for 
a  wood-pole  line  are  given  in  the  sample  estimate  which  follows; 
but  cost  handbooks  or  manufacturers'  catalogues  should  be  con- 
sulted for  specific  information  of  this  kind. 

18.  Cost  of  Insulators. — The  cost  of  insulators  increases 
rapidly  with  the  rise  of  the  working  voltage.  The  curve  of  Fig. 
17  gives  approximate  average  prices  of  insulators  complete  with 
pins  or  suspension  links  for  pressures  up  to  140,000  volts.  The 
prices  are  per  insulator  or  per  series  of  insulator  units.  The 


$15.00 


3.00 
2.00 
1.00 


10    20     30     40     50     60     70    80     90.    10Q_UO   120  130_140 
Line  Pressure  in -Kilo-volts 

FIG.  17. — Approximate  cost  of  line  insulators. 


suspension  type  of  insulator  consisting  of  a  number  of  units  in 
series  is  almost  universally  used  for  pressures  exceeding  60,000 
volts.  One  golden  rule  which  applies  to  all  overhead  transmis- 
sions is  that  it  is  false  economy  to  reduce  first  cost  by  putting 
in  cheap  and  possibly  unreliable  insulators. 

19.  Duplicate  Lines. — A  point  of  great  importance  in  connec- 
tion with  power  transmission  undertakings  is  the  means  adopted 
to  guard  against  interruption  of  supply.  If  it  is  allowed  that 
the  least  reliable  part  of  a  transmission  system  is  the  line  itself, 
it  is  certainly  advisable,  when  circumstances  permit,  to  dupli- 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    43 

cate  the  lines,  the  two  sets  of  conductors  being  connected  in 
parallel,  if  desirable,  under  normal  conditions.  The  best  pro- 
tection against  interruption  would  be  afforded  by  carrying  the 
two  sets  of  conductors  on  separate  poles,  preferably  by  different 
routes;  but  this  would  almost  double  the  cost  of  the  line,  and  it 
is  usual  to  carry  duplicate  lines  either  on  one  set  of  poles,  or  on 
two  sets  of  poles  erected  side  by  side  on  the  same  right  of  way. 
As  an  alternative  to  the  duplication  of  the  lines,  the  provision 
of  reserve  generating  plant  at  the  receiving  end  may  be  consid- 
ered, and  a  comparison  should  be  made  between  the  relative 
advantages  and  costs  of  the  various  alternatives. 

A  good  example  of  steam-driven  auxiliary  plant  in  connection 
with  hydro-electric  power  stations,  is  the  oil-burning  steam- 
generating  station  of  the  Southern  California  Edison  Co.,  situ- 
ated 25  miles  from  the  city  of  Los  Angeles  and  capable  of  con- 
necting in  parallel  with  the  60,000-volt  and  30,000-volt  systems 
ordinarily  supplied  by  the  Kern  River  and  other  hydro-electric 
generating  stations  of  this  company. 

20.  Costs  of  Typical  Transmission  Lines. — It  would  be  pos- 
sible to  give  a  large  number  of  figures  relating  to  material  and 
labor  costs  of  completed  transmission  lines;  but  the  conditions 
of  transport  of  materials  and  quality  of  labor  differ  widely,  and 
without  complete  knowledge  of  these  conditions,  such  figures 
are  liable  to  be  misleading.  For  this  reason  two  ideal  pre- 
liminary estimates,  one  referring  to  a  wood  pole  line,  and  the 
other  to  a  steel  h'ne,  are  here  reproduced,  in  the  hope  that  they 
may  be  useful  as  a  basis  on  which  somewhat  similar  estimates 
may  be  shaped. 

PRELIMINARY  ESTIMATE  No.  1 

Wood  pole  transmission  line,  20  miles  long,  carrying  one  three- 
phase  line.  Line  pressure  22,000  volts.  Span  130  ft.  There 
is  no  grounded  overhead  guard  wire;  but  two  telephone  wires 
are  carried  on  the  same  set  of  poles.  An  allowance  of  20  per 
cent,  is  made  for  extra  insulators  and  fixtures  to  permit  of  dou- 
bling these  on  corner  poles  and  in  other  selected  positions. 

PRELIMINARY  ESTIMATE  No.  2 

''Flexible"  type  steel  tower  line,  60  miles  long,  with  two  sets 
of  three-phase  conductors.  Line  pressure  =  80,000  volts. 


44 


ELECTRIC  POWER  TRANSMISSION 


Average  span,  480  ft.  Spacing  between  wires,  8%  ft.  A 
Siemens-Martin  steel  cable,  acting  as  grounded  guard  wire, 
joins  the  tops  of  all  towers.  Insulators  of  the  suspension  type. 
No  telephone  wires.  Cost  of  right-of-way  not  included  in 
estimate. 


PRELIMINARY  ESTIMATE  No.  1 

MATERIALS  (EXCLUDING  CONDUCTORS) 

40  creosoted  cedar  poles,  35  ft.  long,  Sin. 

diam.  at  top $400.00 

48  cross-arms,  3^  by  4^  in.  by  4  ft.  long .  .       20 . 00 
96  galvanized-iron  braces,  1%  by  K  by  28 

in.  long 7 . 00 

32  galvanized  bolts,   %  by  12K  in.,   with 

washers 

8  galvanized    bolts,    %    by    16    in.,    with 

washers 

1 6  galvanized  spacing  rods,  %  by  1 6  in .,  with        6 . 50 

nuts  and  washers 

48  galvanized  lag  screws,  %  by  3J^  in 

96  galvanized    carriage    bolts,     %    by    4J£ 

in 

1500  ft.  galvanized  7-strand  ^{c-in.  guy  wire. .       12. 00 
12  anchor  rods  with  nuts  and  washers  and 

necessary  timber  for  anchor  logs 7  00 

24  galvanized  guy  clamps  with  bolts 3 . 00 

8  galvanized   sheet-iron  bands   to  prevent 

cutting  of  poles  by  guy  wire 0. 50 

12  standard  thimbles  for  guy  wire 0. 50 

20  galvanized-iron  lightning  conductors,  with 

bolts 5.50 

20  ground  plates  or  galvanized-iron  pipes ...         8 . 00 
Staples  and  sundries,  including  allowance 

for  breakages  and  contingencies 20 . 00 

80  telephone  wire  insulators  (glass) j 

80  side  brackets  for  same  (wood);  5-in.  wire    \  10.00 

nails J 

144  H.T.  porcelain  insulators 42 . 00 

96  galvanized-iron     insulator     pins     with 

porcelain  bases 16 . 00 

48  special  pole-top  insulator  pins,  with  bolts .       21 . 00 


Total  material  cost  per  mile  of  line . 


$579.00 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    45 

LABOR 

Clearing  50  ft.  on  each  side  of  pole  line  @  $30 

per  acre 363 . 00 

Distributing  poles  and  other  materials  along  the 

line ; 50.00 

Trimming  poles,  cutting  gains,  drilling  holes, 

setting  cross-arms 40. 00 

Digging  holes  and  erecting  poles,  including  the 

necessary  guying 100. 00 

Fixing  insulators  and  stringing  wires,  including 

telephone  line 100 . 00 

Supervision  and  sundry  small  labor  items 40. 00 

Loss  and  depreciation  of  tools 15 . 00 

Management  and  preliminary  engineering 

work 35.00 


Total  cost  per  mile  for  charges  other  than 

materials $743 . 00 

Total  cost  per  mile,  excluding  cost  of  con- 
ductor material $1322 . 00 

CONDUCTORS 

16,000  ft.  No.  1  copper  conductors  (hard-drawn);  700  ft.  No.  4 
copper  for  ties  (soft);  10,800  ft.  No.  10  copper  for  telephone 
circuit;  4550  Ib.  @  $20  per  100  Ib 910. 00 


Total  cost  per  mile  of  finished  line  not  including  interest  on 

capital  invested  during  construction  period $2232 . 00 

PRELIMINARY  ESTIMATE  No.  2 
MATERIALS  (EXCLUDING  CONDUCTORS) 

10  flexible  type,   galvanized-steel,    A-frame 

towers  @  $90 $900. 00 

1  galvanized-steel  strain  tower 170. 00 

concrete  foundations  where  necessary ....       80 . 00 
5600ft.  %6-in.  galvanized  Siemens- Martin  steel 
strand  cable  for  guard  wire  and  head 

guys  on  half-mile  flexible  towers 130 . 00 

4  anchor  rods,  complete  with  clamps  and 

thimbles  for  guy  wire 4 . 00 

90  sets  of  suspension-type  insulators,  includ- 
ing strain  insulators  and  small  allow- 
ance for  breakages,  complete  with 

clamps 500.00 

Sundry  small  items  or  special  material ...       50 . 00 


Total  material  cost  per  mile  of  line $1834 . 00 


46  ELECTRIC  POWER  TRANSMISSION 

LABOR 

Clearing  60  ft.  on  each  side  of  line  at  average 

cost  of  $25  per  acre $363 . 00 

Distributing  towers  and  other  materials  along 

the  line 90.00 

Foundations  for  towers 85 . 00 

Assembly  of  parts  and  erection  of  towers 150 . 00 

Fixing  insulators  and  stringing  wires 160.00 

Supervision  and  sundry  small  labor  items 50 . 00 

Allowance  for  loss  and  depreciation  of  tools ...  20 . 00 
Allowance   for   management   and   preliminary 

engineering  work 50 . 00 


Total  charges  other  than  materials,   per 
mile...  968.00 


Total    cost    per    mile    not    including    con- 
ductor material $2802 . 00 

CONDUCTORS 

No.  00,  hard-drawn,  stranded-copper  conductors;  small  amount 
of  No.  2  soft  copper  for  occasional  ties;  special  clamps,  shields, 
jointing  materials,  etc. ;  13,350  Ib.  @  $20  per  100  Ib $2670 . 00 


Total  cost  per  mile  of  finished  line,  not  including  right-of- 
way  or  interest  on  capital  invested  during  construction . . .    $5478 . 00 

The  curves  of  Fig.  18  are  intended  to  supplement  the  figures 
of  the  typical  estimates.  They  give  approximate  costs  of  trans- 
mission poles  or  towers,  with  insulators  fixed  in  position.  These 
costs  are  the  averages  of  many  actual  figures,  and  give  an  approxi- 
mate idea  of  the  total  expenditure  per  mile  of  line  for  various 
voltages;  they  do  not  include  any  clearing  that  may  be  necessary 
in  wooded  country,  or  payments  for  right-of-way.  It  is  assumed 
that  the  conductors  are  of  average  size  (No.  000  B.  &  S.  gauge), 
but  the  actual  cost  of  the  conductors,  whatever  the  size,  must  be 
added  to  the  costs  indicated  by  the  curves  in  order  to  arrive  at  the 
total  cost  of  the  finished  line.  These  curves,  however,  do  in- 
clude an  amount  to  cover  the  labor  cost  of  stringing  the  wires, 
which  will  generally  lie  between  $25  and  $75  per  wire,  per  mile, 
depending  upon  the  size  and  number  of  the  wires  and  the  nature 
of  the  ground  covered  by  the  transmission  line.  The  lower  curve 
of  Fig.  18  refers  to  wood  poles  or  rigid  steel  towers  (for  the  higher 
voltages)  carrying  three  conductors;  while  the  upper  curve  refers 
to  a  single  set  of  poles  or  towers  carrying  six  conductors.  It 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    47 

should  be  understood  that  the  curves  of  Fig.  18  give  only  an 
approximate  indication  of  the  probable  capital  expenditure  on 
the  line.  The  actual  cost  will  depend  upon  the  character  of  the 
country,  the  nature  of  the  ground,  and  other  local  conditions, 
such  as  cost  of  labor  and  facilities  for  transportation.  These, 
together  with  the  weather  conditions,  force  of  wind  and  possible 
loading  of  wires  with  sleet  or  ice,  will  determine  the  most  eco- 
nomical span  and  the  average  height  of  pole  or  tower.  The  cost, 
as  previously  mentioned,  will  also  depend  upon  the  material  of 
the  conductors,  as  a  larger  or  smaller  sag  will  influence  spans 


>44UU 

/ 

3000 

/ 

/ 

;ff 

/ 

, 

^; 

y 

/ 

2100 

2200 
2000 
1800 
1000 

1200 
1000 
800 
600 
400 
200 

^ 

. 

/ 

^ 

X 

/ 

*/ 

/ 

/ 

/ 

> 

^ 

y 

: 

^ 

#& 

'>• 

^ 

^ 

^ 

"** 

These  curves  give  the  approocim 
cost  per  taile  of  a  transmission 
line  complete  with  insulators, 
Vot  Including  the  cost  of  condu 
right  of  way,  clearing  -around  1 
wooded  country*  01  Interest  on 
capital  during  con  t  ruction  perl 
The  labor,  cost  of  springing  the 
conductors  iff  included. 

ate 

^ 

^ 

bus 
ctors, 

0      |  

>a.  r~ 

*•**" 

40          50         60          70 
Pressure,  Kilo-volts 

FIG.   18. — Approximate  cost  of  overhead  transmission  lines. 

and  height  of  poles.  The  weight  and  diameter  of  conductors, 
by  affecting  the  required  strength  of  the  supports,  will  be  factors 
in  determining  the  cost  of  the  complete  line,  apart  from  any 
difference  in  the  value  of  the  conductors  themselves.  The  actual 
cost  of  stringing  very  light  or  very  heavy  conductors  will  also 
differ  from  the  average  amount  allowed  for  the  purpose  of  plot- 
ting the  curves.  The  number  and  style  of  lightning  conductors, 
if  any,  and  whether  or  not  one  or  more  grounded  guard  wires 
are  strung  above  the  conductors  will  obviously  modify  the  average 
figures.  Although  steel  poles,  or  steel  towers,  will  generally  be 
found  more  economical  than  a  wood  pole  line  for  voltages  above 
44,000  on  account  of  the  heavier  insulators,  wider  spacing  be- 
tween conductors,  and  generally  greater  height  of  support,  it 


48  ELECTRIC  POWER  TRANSMISSION 

does  not  follow  that  wood  poles  or  wood-pole  structures  may  not 
prove  economical,  even  for  comparatively  high  voltages,  in  coun- 
tries where  suitable  timber  is  plentiful  and  the  ready  means  of 
transportation  and  erection  of  steel  towers  are  wanting.  For 
instance,  the  cost  per  mile,  as  obtained  from  Fig.  18,  is  $2600 
for  a  single  100,000-volt  three-phase  line  supported  on  rigid  square 
base  steel  towers,  and  it  is  not  improbable  that  an  entirely  satis- 
factory wood  pole  line  could  be  built  under  favorable  conditions 
at  a  figure  appreciably  below  $2000  per  mile  (not  including 
conductors,  right-of-way,  or  clearing  wooded  land).  But  it  does 
not  follow  that  the  wood  pole  line  is  the  most  economical,  because 
the  probable  cost  of  maintenance,  repairs,  replacements  of  de- 
cayed or  damaged  poles,  and  all  charges  to  be  met  annually  or 
periodically  during  future  years  must  be  carefully  considered 
before  a  final  decision  can  be  arrived  at. 

21.  Cost  of  Overhead  Conductors. — The  capital  expenditure 
on  conductors  will  depend  upon  the  material  and  the  total  weight. 
It  is  not  proposed  to  discuss,  in  this  place,  the  relative  merits  of 
copper  and  aluminum  as  conductor  materials,  but  it  may  be  well 
to  point  out  that  although  the  market  values  of  these  metals 
may  be  such  that  the  use  of  aluminum  may  lead  to  some  saving 
on  first  cost,  there  are  many  engineering  points  to  be  most  care- 
fully considered  before  definitely  adopting  either  metal.  The 
weight  of  the  conductors  necessary  to  transmit  a  certain  amount 
of  power  over  a  definite  distance  will  obviously  depend  upon  the 
voltage,  but  apart  from  the  engineering  difficulties  encountered 
at  the  higher  voltages,  there  are  economic  considerations  which 
determine  the  maximum  voltage  suitable  for  any  given  conditions. 
Among  these  may  be  mentioned  a  possible  increase  in  the  cost 
of  generating  plant  for  the  higher  pressures,  the  greater  cost  of 
step-up  and  step-down  transformers  and  of  the  control  apparatus, 
together  with  the  line  insulators,  entering  bushings,  etc.  The 
transmission  line  poles  or  towers  will  also,  as  previously  men- 
tioned, cost  more  for  the  higher  pressures,  because  of  the  wider 
spacing  between  conductors  and  the  greater  length  of  insulator 
string.  Then  again,  with  the  extra  high  pressures,  the  increased 
losses  through  leakage  over  insulators  and  possible  corona  losses 
may  be  quite  appreciable. 

Given  a  definite  amount  of  power  to  be  transmitted,  and  a 
definite  line  pressure,  the  current  can  be  calculated ;  and  the  eco- 
nomic conductor  cross-section — and  therefore  the  weight  and 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    49 

cost  of  the  conductors — will  be  directly  proportional  to  this 
current.  It  is  only  of  recent  years  that  this  fact  appears  to  have 
been  generally  recognized,  and  yet,  so  long  ago  as  1885,  in  his 
Cantor  lectures  delivered  in  London,  Prof.  George  Forbes  said: 
''The  most  economical  section  of  conductor  is  independent  of 
e.m.f.  and  distance,  and  is  proportional  to  the  current."  The 
determination  of  the  current  value  to  be  used  in  the  calculation  of 
conductor  sections  is  a  real  difficulty.  It  must  not  be  supposed 
that  even  a  knowledge  of  the  load  factor  is  sufficient  by  itself. 
The  load  factor,  being  the  ratio  of  average  load  to  maximum  load, 
does  not  give  the  relation  between  the  average  PR  loss  and  the 
I2R  loss  of  maximum  output.  The  power  lost  in  the  conductors 
of  a  constant  potential  supply  is  proportional  to  the  square  of 
the  power  transmitted.  On  the  basis  of  the  average  hydro- 
electric load  curve,  if  the  load  factor  is  50  per  cent.,  the  load  on 
which  the  average  transmission  line  losses  should  be  based — 
being  the  square  root  of  the  mean  of  the  square  of  the  power 
transmitted- — will  probably  be  found  to  be  more  nearly  60  per 
cent,  than  50  per  cent,  of  the  maximum  load. 

Although  the  discussion  which  follows  refers  mainly  to  over- 
head transmission  lines,  the  same  general  principles  should 
govern  the  choice  of  conductor  cross-section  in  underground 
cables.  As  an  economic  problem,  the  underground  system  of 
transmission  differs  from  the  overhead  system  mainly  in  the 
fact  that  the  cost  of  the  insulation  in  a  cable  is  a  function  of  the 
conductor  diameter,  whereas  the  cost  of  line  insulators  is  less 
closely  connected  with  the  size  or  weight  of  the  overhead  con- 
ductors. The  cost  of  insulation  is  relatively  more  important  in 
cable  systems  than  in  overhead  transmissions.  Some  reference 
to  the  economics  of  power  transmission  by  underground  cables 
will  be  made  in  Chapter  VII. 

22.  Economic  Size  of  Conductor.  Kelvin's  Law.— Before 
considering  to  how  great  an  extent  the  voltage  may  be  raised,  in 
order  to  keep  down  the  current,  without  exceeding  the  limits 
determined  by  economic  considerations,  it  will  be  well  to  examine, 
in  some  detail,  the  fundamental  principle  known  as  Kelvin's  law, 
by  which  the  proper  size  of  conductor  to  carry  a  known  current 
is  determined.  In  this  connection  it  is  of  no  consequence  whether 
the  transmission  is  by  direct  or  alternating  currents,  single 
phase  or  polyphase.  If  conductors  have  to  be  provided  to 
carry  a  current  of  known  amount,  these  may  be  of  large  cross- 


50  ELECTRIC  POWER  TRANSMISSION 

section  and  therefore  of  high  initial  cost,  but  of  so  low  a  resistance 
that  the  PR  losses  will  be  small;  or  they  may  be  of  small  cross- 
section  and  high  resistance,  the  capital  expenditure  on  which 
will  be  small;  but  in  which  the  I*R  losses  will  be  large.  The 
economical  size  of  conductor  for  any  given  transmission  will  there- 
fore depend  on  the  cost  of  the  conductor  material  and  the  cost 
of  the  power  wasted  in  transmission  losses;  and  the  law  of  maxi- 
mum economy  may  be  stated  as  follows:  The  annual  cost  of  the 
energy  wasted  per  mile  of  the  transmission  line,  added  to  the  an- 
nual allowance  (per  mile)  for  depreciation  and  interest  on  first  cost, 
shall  be  a  minimum. 

If  it  is  assumed  that  the  cost  of  poles  or  towers,  insulators  and 
other  materials  (apart  from  the  conductors  themselves)  including 
the  labor  on  erection  and  stringing  of  wires,  is  independent  of 
the  actual  size  of  conductor,  then  the  only  variable  item  in  the 
capital  expenditure  is  directly  proportional  to  the  cross-section 
(or  weight)  of  the  conductor,  and  since  the  PR  losses  (for  a  given 
current)  are  inversely  proportional  to  the  conductor  cross-section, 
the  law  of  maximum  economy  is  greatly  simplified,  and  in  fact 
becomes  Kelvin's  law,  which  may  be  expressed  as  follows: 

The  most  economical  section  of  a  conductor  is  that  which  makes 
the  annual  cost  of  the  IZR  losses  equal  to  the  annual  interest  on  the 
capital  cost  of  the  conductor  material,  plus  the  necessary  annual 
allowance  for  depreciation.  The  cross-section  should,  therefore, 
be  determined  solely  by  the  current  which  the  conductor  has  to 
carry,  and  not  by  the  length  of  the  line  or  an  arbitrary  limit  of 
the  percentage  full-load  pressure  drop.  If  there  are  reasons 
which  make  a  large  pressure  drop  undesirable,  then,  if  necessary, 
economy  must  be  sacrificed,  and  the  line  calculated  on  the  basis 
of  regulation  only.  It  will,  however,  generally  be  found  that 
the  economic  conductor  will  give  reasonably  good  regulation. 

The  diagram,  Fig.  19,  shows  clearly  how  the  minimum  total 
annual  cost  occurs  when  the  cost  per  annum  of  the  wasted  energy 
is  equal  to  the  capital  cost  expressed  as  an  annual  charge;  and  if 
desired  a  graphical  solution  of  Kelvin's  law  can  readily  be  ob- 
tained by  this  means.  In  Fig.  19,  the  horizontal  distances  meas- 
ured to  the  right  of  the  point  O  represent  increasing  conductor 
resistances;  while  the  vertical  distances  represent  money.  The 
curve  A  shows  how  the  annual  charges  depending  on  capital 
outlay  decrease  with  increase  of  conductor  resistance;  while  the 
straight  line  B  indicates  the  growth  of  the  cost  of  wasted  power; 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    51 

this  being  directly  proportional  to  the  resistance.  By  adding 
the  ordinates  of  curves  A  and  B,  the  curve  C  is  obtained,  of  which 
the  lowest  point  indicates  the  resistance  per  mile  of  conductor 
which  will  be  the  most  economical  to  use,  whatever  may  be  the 
length  of  the  line,  or  the  pressure  required  at  the  receiving  end. 


.6  1.0 

Resistance,  Ohms  per  Mile,  of  Single  Conductor 

FIG.  19. — Graphical  illustration  of  Kelvin's  law. 


It  will  be  observed  that  this  minimum  occurs  where  the  two 
curves  cross. 

23.  Practical  Method  of  Applying  Kelvin's  Law.— The  follow- 
ing formulas  have  been  evolved  with  a  view  to  facilitating 
the  calculation  of  conductor  sizes  to  give  the  most  economical 


52  ELECTRIC  POWER  TRANSMISSION 

results  on  overhead  transmissions.  In  every  case  the  lesser 
factors  which  may,  to  a  small  extent,  influence  the  results  of  the 
problem  will  be  disregarded,  but  they  may  be  taken  into  account 
when  the  final  details  of  the  transmission  line  are  being  con- 
sidered. On  the  other  hand,  it  will  generally  be  found  that  the 
application  of  Kelvin's  law  in  its  simplicity,  without  regard  to 
such  influences  as  the  possible  variations  in  cost  of  supports,  in- 
sulators, etc.,  depending  upon  the  size  of  the  conductors,  will 
give  results  sufficiently  accurate  for  practical  purposes,  and  this 
for  two  important  reasons : 

1.  A  small  variation  in  the  diameter  of  the  conductor  either 
on  the  large  or  the  small  side  is  usually  of  very  little  consequence 
from  the  economic  point  of  view. 

2.  As  the  standard  size  of  conductor  nearest  in  diameter  to 
the  theoretically  correct  size  is  generally  selected,  refinements  or 
increased  accuracy  in  the  calculations  will  rarely  affect  the.  size 
of  wire  which  is  ultimately  decided  upon. 

24.  Economic  Ohmic  Voltage  Drop. — It  is  not  generally 
realized  that  when  the  size  of  a  conductor  is  determined  by  the 
application  of  Kelvin's  law  the  ohmic  drop  of  pressure  per  unit 
length  of  conductor  is  independent  of  the  actual  voltage  or  the 
current  to  be  carried,  and  therefore  bears  no  definite  relation  to 
the  total  amount  of  power  to  be  transmitted.  The  economic 
data  and  assumptions  alone  determine  the  ohmic  drop  in  volts  per 
unit  length  of  conductor,  and  this  will  be  a  constant  quantity 
whatever  the  number  of  conductors  or  system  of  electric  trans- 
mission adopted,  the  total  amount  of  power  to  be  transmitted, 
or  the  voltage  ultimately  decided  upon.  This  fact  very  consider- 
ably simplifies  the  problem  in  its  earlier  stages. 

The  formula  for  the  economic  voltage  drop  may  be  arrived 
at  as  follows,  bearing  in  mind  that  the  annual  charges  to  be 
considered  are  (1)  an  annual  charge  for  interest  and  depreciation 
on  the  cost  of  the  line  wire;  (2)  the  annual  cost  of  the  energy 
wasted  in  the  conductor  in  the  form  of  I*R  losses,1  and  that  the 
equality  of  these  two  items  of  cost  determines  the  size  of  the  most 
economical  conductor. 

Annual  Charges  Depending  Upon  Cost  of  Conductor. — Let  p 
be  the  price  to  be  paid  for  100  Ib.  weight  of  conductor  and  a  the 

1  Other  losses  due  to  leakage  over  insulators  and  through  the  air  should 
be  taken  into  account  when  considering  the  choice  of  e.m.f.,  especially  if 
this  should  exceed  60,000  volts. 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    53 

percentage  to  be  taken  to  cover  the  annual  interest  and  deprecia- 
tion, then,  if  R  be  the  resistance  in  ohms  per  mile  of  the  conductor, 

Annual  charge  =  ~  X  pX^XK  (16) 

where  K  is  a  constant  depending  upon  the  material  of  the  con- 
ductor and  the  temperature. 

Annual  Cost  of  Energy  Lost—  Let  p\  be  the  cost  per  kw.-year 
of  the  wasted  energy;  then, 
annual  cost  per  mile  of  conductor  =  p\  Xkw.  lost  per  mile. 


where  Er  stands  for  ohmic  drop  in  volts  per  mile  of  conductor. 
In  order  to  satisfy  the  condition  of  equality  between  the  values 
(16)  and  (17)  we  must  write 

a  X  p  X  K  =   Pl  X  Erz 
100  X  R         1000  X  R 
whence 

Er*=WK^^  (18) 

If  the  temperature  is  about  20  degrees  Centigrade,  and  the 
material  of  the  line  is  copper,  the  constant  K  may  be  taken  as 
8.76,  while  for  aluminum  it  works  out  at  4.32.  Inserting  these 
values  in  the  last  formula,  the  economic  ohmic  voltage  drop  per 
mile  of  copper  conductor  becomes: 


Er=  9.35  ^  (19) 

and  for  aluminum: 

(20) 


If  preferred,  these  formulas  may  be  put  in  the  form: 

Circular  mils  per  ampere  (copper)  =  5800  \l-~-        (21) 

\  a  /\  p 

and, 

Circular  mils  per  ampere  (aluminum)  =  13,400  A/ — -1 —    (22) 

25.  Economic  Voltage,  and  Calculation  of  Conductor  Sizes. — 
Having  ascertained  what  will  be  the  most  economical  ohmic 


54  ELECTRIC  POWER  TRANSMISSION 

drop  of  pressure  per  mile  of  conductor  without  reference  to  the 
total  amount  of  power  to  be  transmitted,  the  size  of  the  conductor 
cannot  be  determined  unless  the  value  of  the  current  is  known, 
and  this  will  depend  upon  the  pressure  at  which  the  energy  will  be 
transmitted. 

If  the  cost  of  the  conductors  forming  the  transmission  line, 
and  the  PR  losses  therein,  were  the  only  considerations,  a  high 
voltage  would  in  all  cases  be  desirable  on  account  of  the  corre- 
sponding reduction  of  current  for  a  given  amount  of  energy  to  be 
transmitted.  But,  apart  from  the  extra  cost  of  the  line  due  to 
the  better  insulation  and  wider  spacing  of  wires  required  by  the 
higher  pressures,  the  cost  of  generation  and  transformation  of 
high-pressure  energy  must  be  taken  into  account,  and  as  the 
extra  cost  per  kilowatt  of  equipment  for  generating  at  high  pres- 
sures will  depend  largely  upon  the  total  output  required,  it 
follows  that  the  most  economical  pressure  will  bear  some  relation 
to  the  total  power  to  be  transmitted.  This  is  apart  from  the 
distance  of  transmission,  which  is  the  most  important  factor 
governing  the  choice  of  voltage.  If  the  distance  is  great  it  is 
obvious  that  the  reduction  of  material  cost  and  power  losses  in 
the  line  due  to  the  employment  of  higher  pressures  will  be  rela- 
tively of  far  greater  importance  than  the  increased  cost  of  plant  in 
generating  and  transforming  stations.  On  the  other  hand, 
the  employment  of  very  high  pressures  even  on  a  comparatively 
long  line  might  not  be  justified  if  the  total  amount  of  power 
to  be  transmitted  were  very  small. 

As  a  first  approximation,  the  writer  has  found  the  following 
formula  useful  in  getting  out  preliminary  estimates;  the  line 
voltage  given  by  the  formula  agrees  generally  with  modern 
practice. 

I         k.w. 
Line  pressure  (kilovolts)  =  5.5  -\/L  +  -^~  (23) 

This  empirical  formula  may  be  used  for  estimating  the  probable 
economical  transmission  voltage  on  lines  over  20  miles  in  length. 
The  symbol  L  stands  for  the  distance  of  transmission  in  miles, 
while  k.w.  stands  for  the  estimated  maximum  number  of  kilo- 
watts that  will  have  to  be  transmitted  over  one  pole-  or  tower- 
line. 

Given  the  amount  of  power  to  be  transmitted  and  the  length 
of  line,  one  can  with  the  aid  of  formula  (23)  decide  upon  a  stand- 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    55 

ard  voltage  and  proceed  with  the  calculations  for  current  and 
size  of  conductor;  but  it  is  necessary  always  to  bear  in  mind  that 
a  transmission  line  cannot  be  considered  by  itself;  it  must  be 
treated  as  part  of  a  complete  scheme  of  transmission  and  dis- 
tribution, and  the  best  voltage  to  use  on  any  given  system  can 
generally  be  arrived  at  only  by  a  method  of  trial  and  error,  taking 
into  account  the  cost  of  the  various  parts  of  the  complete  system 
as  influenced  by  alterations  in  the  transmission  voltage.  No 
accurate  formula  can  be  evolved  which  would  be  applicable  to 
all  the  varied  conditions  encountered  in  actual  work;  but  a 
practical  method  of  attaining  the  required  end  will  be  explained 
later. 

26.  Example  Illustrating  Quick  Method  of  Determining  Eco- 
nomic Size  of  Conductors. — For  the  purpose  of  working  out  a 
practical  example  the  following  assumptions  have  been  made : 

Total  kilowatts  to  be  transmitted,  P  =  12,000. 

System,  three-phase. 

Power-factor  =  0.8. 

Distance  of  transmission  =  120  miles. 

Copper  conductors  to  be  used,  the  cost  p  being  $20  per  100 
Ib. 

Percentage  to  be  taken  to  cover  depreciation  and  annual 
interest  on  cost  of  copper,  a  =  12.5. 

Estimated  cost  of  wasted  power  per  kilowatt-year,  pi  =  $22. 

The  economic  voltage  drop  per  mile  of  single  conductor  will  be, 
by  formula  (19), 

P     _Q«       I12'5    ><    20 

tir  =  \).65  Aj 22 — 

=  31.5  volts 
The  transmission  voltage  as  given  by  formula  (23)  is: 


Kilo  volts  =  5.5  Jl20  + 

=  86 

or,  say,  88,000  volts  at  the  receiving  end. 
The  current  per  conductor  will  be: 
=  Watts 


A/3  X  E  X  cos  e 

12,000,000 

:  A/3  X88,000  X  08 
=  98.4  amp. 


56  ELECTRIC  POWER  TRANSMISSION 

J7t  Q  1        C 

Resistance    of    conductor    per    mile  =  -y  =  T^-T  =  0.32     ohm, 

/      yoiTc 


and  since  No.  3-0  B.  &  S.  wire  has  a  resistance  of  0.326  ohm 
per  mile,  that  is  the  standard  size  which  should  be  adopted 
unless  a  more  careful  study  of  the  complete  scheme  should  lead 
to  a  different  decision  in  regard  to  the  pressure  of  transmission. 
Since,  for  a  given  amount  of  power  to  be  transmitted,  the 
current  will  vary  inversely  as  the  pressure,  it  follows  that  the 
resistance  per  mile  of  conductor  to  give  the  economic  voltage 
drop  per  mile  (31.5  volts  in  this  particular  example)  will  be 
directly  proportional  to  the  pressure  at  which  the  power  is 
transmitted.  Thus  if  110,000  volts  were  found  to  be  a  more 
economical  pressure  than  88,000,  the  ohms  per  mile  of  conductor 

would  be  -  -  =  0.4,    the    nearest    standard   size  being 

00 

No.  2-0  (ohms  per  mile  =  0.41). 

Power  Lost  in  Line.  —  If  w  stands  for  the  total  IZR  watts  lost 
in  the  three  conductors,  based  on  the  calculated  value  of  the 
resistance,  then 

w  =  3  X  length  of  line  X  /  X  Er 
=  3  X  120  X  98.4  X  31.5 
=  1115  k.w. 

on  the  assumption  that  a  transmission  pressure  of  88,000  volts 
is  adopted;  and  since  the  total  kilowatts  transmitted  are  12,000, 
the  percentage  power  loss  is  : 

1115  X  100 

12,000         :  9-3  Per  cent. 

Voltage  Regulation.  —  The  drop  in  pressure  per  conductor, 
due  to  ohmic  resistance  only,  will  be: 

Er  X  length  of  line  =  31.5  X  120  =  3780  volts 
or  3780  X  \/3  =  6550  volts  between  wires,  since  the  system  is 
three-phase  and  the  volts  Er  refer  to  a  single  conductor  only. 
The  percentage  ohmic  drop  is,  therefore: 

6550 
OQ     =  7.44  per  cent. 

oo 

This  figure  alone  does  not,  however,  give  much  indication 
as  to  what  will  be  the  actual  regulation  of  the  line,  as  the  effects 
of  inductance  and  electrostatic  capacity  must  be  taken  into 
account  and  the  resultant  difference  of  pressure  between  the 
transmitting  and  receiving  ends  of  the  line  calculated  by  any 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    57 

one  of  the  usual  methods.  The  resultant  pressure  drop  may 
be  found  to  be  excessive;  it  may  be  such  as  cannot  readily  be 
dealt  with  in  a  practical  scheme,  and  in  such  a  case  the  economy 
of  the  line  may  have  to  be  sacrificed  by  putting  in  larger 
conductors. 

It  is  obvious  that  other  conditions  may  render  it  inexpedient 
or  impossible  to  adopt  the  most  economical  size  of  conductor 
as  calculated  by  the  application  of  Kelvin's  law,  but  in  such 
cases  experience  and  common  sense  will  usually  indicate  the 
proper  course  to  follow.  If  the  economic  size  of  wire  is  small 
it  is  possible,  but  not  probable,  that  there  may  be  trouble  due 
to  excessive  heating.  A  want  of  mechanical  strength,  or  loss 
of  power  due  to  corona  formation,  are  more  likely  to  lead  to  the 
selection  of  a  conductor  diameter  larger  than  the  "economic 
size."  If,  on  the  other  hand,  the  conductor  diameter  is  very 
large,  there  may  be  difficulties  in  handling  and  in  taking  the  strain 
on  the  individual  insulators.  The  remedy  in  this  case  is  obviously 
to  subdivide  the  single  circuit  into  two  or  more  parallel  circuits, 
and,  in  fact,  there  are  many  advantages  in  doing  so  rather  than 
running  very  heavy  single  conductors.  One  particular  aspect  of 
the  question  of  subdivision  of  transmission  lines  is  dealt  with  in 
Appendix  I. 

Again,  even  from  the  economic  point  of  view,  the  case  might 
arise  of  a  temporary  installation  intended  to  give  a  quick  return 
on  capital  invested,  and  an  exceptionally  small  size  of  wire  giv- 
ing a  large  IZR  loss  might  produce  the  best  results.  This,  how- 
ever, leads  to  the  consideration  of  the  most  important  factor 
in  the  whole  problem,  namely,  the  correctness  of  the  estimates 
of  costs,  depreciation  allowances,  and  power  transmitted,  upon 
which  the  value  of  the  calculated  results  will  mainly  depend. 
It  is  here  that  the  experience,  foresight  and  sound  judgment  of 
the  engineer  must  necessarily  play  an  important  part,  and  it 
is  not  possible  in  this  chapter  to  do  more  than  draw  attention  to 
some  considerations  which  must  not  be  overlooked. 

27.  Estimation  of  Amount  and  Cost  of  Energy  Wasted  in 
Conductors. — The  correct  value  of  the  power  (P)  from  which 
the  value  of  the  current  (/)  is  determined  is  frequently  very 
difficult  to  estimate.  This  is  a  point  which  is  best  considered 
when  determining  the  cost  of  the  wasted  energy.  It  is,  however, 
clear  that  the  annual  amount  of  energy  wasted  will  depend 
not  only  on  the  average  value  of  72,  but  also  on  the  time  during 


58  ELECTRIC  POWER  TRANSMISSION 

which  the  average  amount  of  power  may  be  considered  as  being 
transmitted  by  the  wires.  If,  therefore,  it  is  desired  to  estimate 
accurately  the  amount  of  energy  wasted  annually  in  the  lines,  a 
probable  load  curve  for  the  year  should  be  drawn  and  the  aver- 
age Iz  calculated  therefrom.  This  will  give  a  value  for  I  which, 
if  considered  as  flowing  in  the  wires  continuously  throughout  the 
year,  will  lead  to  a  certain  watt-hour  or  yearly  energy  loss, 
the  cost  of  which  it  is  desired  to  know. 

Now,  the  annual  cost  of  production  of  an  additional  elec- 
trical horse-power,  considered  apart  from  the  total  cost  of  pro- 
duction, is  always  'difficult,  if  not  impossible,  to  estimate  ac- 
curately, but  where  coal  is  the  source  of  energy  there  is  at  least 
the  extra  cost  of  coal  consumed  to  be  taken  into  account  when 
estimating  the  production  cost  of  the  lost  energy.  The  case 
is  different  in  a  water-power  generating  station,  where  the  cost 
of  running  the  station  at  full  output  is  very  little  in  excess  of 
the  cost  of  running  at  one-quarter  or  one-tenth  of  maximum 
output,  and  it  is  even  more  difficult  to  decide  upon  a  figure 
which  shall  represent  the  cost  of  wasted  energy  (pi  in  the  cal- 
culations) with  sufficient  accuracy  to  make  the  calculations 
of  the  economic  conductor  of  real  practical  value. 

There  are  two  points  in  connection  with  water-power  proposi- 
tions which  must  never  be  lost  sight  of: 

(1)  If  the  amount  of  water-power  available  is  limited,  while  the 
demand  for  power  is  unlimited,   the   cost  (pi),  of  the  wasted 
energy  may  be  taken  at  the  price  which  the  user  would  actually 
be  prepared  to  pay  for  it  were  it  available  for  useful  purposes. 

(2)  If  the  water-power  is  unlimited  as  compared  with  the  de- 
mand for  power,  the  cost  of  wasted  energy  is  practically  nil,  ex- 
cept for  the  fact  that  a  generating  plant  has  to  be  installed  of  a 
somewhat  larger  capacity  than  would  otherwise  be  necessary; 
and  the  works  cost  of  the  wasted  energy  must,  of  course,  include 
a  reasonable  percentage  to  cover  interest  and  depreciation  on 
this  extra  plant. 

28.  Estimation  of  Percentage  to  Cover  Annual  Interest  and 
Depreciation  on  Conductors. — So  far  as  interest  is  concerned, 
if  cash  is  to  be  paid  for  the  conductors,  the  figure  to  he  taken  for 
interest  on  capital  should  be  on  a  par  with  the  expected  per- 
centage profit  on  the  complete  undertaking;  but  if  the  conductors 
are  mortgaged,  it  is  the  annual  amount  of  the  mortgage  which 
should  be  taken. 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    59 

In  regard  to  depreciation,  the  probable  life  of  the  con- 
ductor must  be  estimated,  and  this,  to  a  certain  extent,  may 
depend  upon  the  life  of  the  transmission  line  considered  as  a 
whole. 

29.  Economic  Voltage.— It  should  be  clearly  understood  that 
the  foregoing  articles  deal  only  with  the  determination  of  the 
correct  size  of  Conductors  based  on  certain  assumptions  as  regards 
voltage  and  power  to  be  transmitted.  The  cost  of  generating 
and  transforming  plant  and  buildings,  as  influenced  by  the  volt- 
age, must  be  carefully  considered,  together  with  the  type  and  cost 
of  pole  line,  so  far  as  these  are  influenced  by  the  size  of  the  con- 
ductors. The  character  of  the  country,  too,  will  have  some  bear- 
ing on  the  design  of  the  transmission  line,  and  the  final  choice  of 
voltage  may  depend  to  some  extent  upon  whether  a  wood  pole 
line  with  comparatively  short  spans  and  (preferably)  small 
spacing  between  wires  is  likely  to  be  more  economical  than  a  line 
with  steel  towers  which  will  permit  of  longer  spans  with  wider 
spacing  between  wires.  In  other  words,  the  total  cost  of  the 
whole  undertaking  and  the  total  annual  losses  of  energy  from  all 
sources,  as  influenced  by  any  change  of  voltage,  must  be  con- 
sidered before  the  line  pressure  as  given  by  formula  (23)  can  be 
definitely  adopted  as  being  the  most  economical  for  the  under- 
taking considered  as  a  whole. 

Clearly,  the  choice  of  the  transmission  voltage  is  a  very  im- 
portant matter;  and  since  it  is  possible  to  determine  the  proper 
voltage  on  purely  economic  grounds,  the  use  of  exceptionally  high 
pressures  merely  because  of  theh1  interest  from  an  engineering 
standpoint,  should  be  discouraged.  On  the  other  hand,  it  would 
appear  that  most  transmission-line  troubles  occur  on  lines  work- 
ing at  pressures  between  30,000  and  80,000  volts;  and  an  im- 
portant consideration  to  bear  in  mind  is  that  more  trouble 
may  be  experienced  with  heavy  currents  than  with  high  voltages, 
owing  to  the  more  serious  effects  of  interruptions  or  transient 
disturbances  when  the  current  is  large,  so  that  greater  security 
may  sometimes  be  obtained  by  increasing  the  voltage  with  a 
view  to  reducing  maintenance  and  operating  costs. 

When  figuring  on  the  best  voltage  for  any  particular  scheme, 
the  capital  cost  of  all  works,  buildings,  or  apparatus,  which  is 
liable  to  be  influenced  by  the  transmission-line  pressure,  together 
with  all  operating  and  maintenance  charges  which  may  be  simi- 
larly influenced,  must  be  taken  into  account.  It  will  usually 


60 


ELECTRIC  POWER  TRANSMISSION 


be  found  convenient  to  reduce  all  such  costs  or  differences  of 
cost  to  the  basis  of  annual  charges. 

30.  Costs  Other  Than  Transmission  Line,  Liable  to  be  Influ- 
enced by  Voltage  Variations. — The  cost  of  a  generating  station 
complete  with  all  plant  and  machinery,  but  not  including  trans- 
mission line,  may  be  anything  from  $25  to  $200  per  kilowatt 
installed.  It  will  depend  on  total  output,  that  is,  on  the  size 
of  the  station,  on  location,  and  transport  and  labor  facilities. 
The  cost  of  a  hydro-electric  station  will  depend  on  the  head  of 
water,  the  amount  of  rock  excavation,  the  size  of  dam,  length 
of  tunnels  and  penstocks,  etc. 

The  figures  given  in  the  accompanying  table  are  approximate 
costs  per  kilowatt  (not  including  the  transmission  line)  for  a 
medium-head  hydro-electric  development  suitable  for  a  total 
output  in  the  neighborhood  of  10,000  k.w.  to  be  transmitted  over 


Transm 

ission-line 

voltage 

30,000 

60,000 

100,000 

Hydraulic  works  outside  power-station  buildings.  . 
Power-station  building,  including  excavations  
Receiving-station  building  
Switch-gear  (both  ends)  
Electrolytic  lightning  arresters  

$15.00 
5.00 
1.00 
1.20 
0.34 

$15.00 

5.06 
1.03 
1.35 
0.66 

$15.00 
5.10 
1.05 
1.70 
1.20 

Transformers  (both  ends)  

2.50 

2.90 

3.50 

Generators  and  exciters  

8.00 

8.00 

8.00 

Cables  in  buildings  entering  bushings  etc 

0  40 

0  40 

0.50 

Crane,  sundries,  and  accessories,  including  pre- 
liminary work                                                   

2.00 

2.10 

2.20 

Turbines  and  hydraulic  equipment  

10.00 

10.00 

10.00 

Total  cost  per  kilowatt  

$45.44 

$46.50 

$48.25 

two  outgoing  three-phase  feeders.  The  usefulness  of  these  figures 
lies  mainly  in  the  indication  they  give  of  the  probable  differences 
in  cost  with  the  variation  of  transmission-line  pressure. 

31.  Annual  Charges  Depending  on  Voltage. — These  charges 
may  be  summarized  as  follows: 

1.  A  percentage  on  all  capital  expenditure,  whether  for  gener- 
ating station,  transmission  line,  or  receiving  stations,  which  is 
not  constant  irrespective  of  voltage. 

2.  The  yearly  cost  of  the  power  lost  in  the  transmission  line. 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    61 

3.  The  yearly  cost  of  power  lost  in  generators  and  transformers 
(the  efficiency  of  the  electrical  plant  will  not  necessarily  be  the 
same  for  all  voltages). 

4.  The  yearly  cost  of  maintenance  and  operation.     This  may 
depend  upon  length  of  spans  in  transmission  line,  and  on  the 
necessary  plant,  switch-gear,  etc.,  to  be  attended  to,  and  kept 
in  working  order. 

The  percentages  referred  to  under  item  (1)  must  include  in- 
terest on  capital  invested  and  depreciation. 

32.  Depreciation. — Depreciation  is  the  loss  of  value  or  com- 
mercial utility  due  to  deterioration  with  age.  The  term  may 
be  used  to  cover  loss  of  value  resulting  from  very  different  causes. 
A  distinction  should  be  made  between  natural  and  functional 
depreciation. 

Natural  or  physical  depreciation  is  the  loss  of  value  due  to 
physical  or  chemical  changes  which,  in  time,  will  render  the 
machine  or  plant  practically,  useless.  Atmospheric  changes, 
alternations  of  heat  and  cold,  wear  and  tear,  erosion,  rust, 
decay,  electrolysis,  are  causes  of  natural  depreciation. 

Functional  depreciation  is  the  loss  of  value  due  to  the  fact 
that,  with  the  lapse  of  time,  the  machine,  plant,  or  structure 
under  consideration  does  not  function  as  efficiently  as  when  it 
was  first  put  into  use,  or  as  efficiently  as  it  should  function 
to  compete  with  improved  methods  or  apparatus.  It  may  be- 
come inadequate  owing  to  rapid  growth  in  the  demand  for  the 
service  which  it  is  intended  to  render;  or  it  may  become  obsolete. 
Thus,  functional  depreciation  may  be  due  to  either  inadequacy 
or  to  obsolescence.  A  machine  or  structure  becomes  obsolete 
owing  to  scientific  or  artistic  developments,  i.e.,  inventions.  It  is 
practically  impossible  to  predict  a  future  state  of  development  in 
any  branch  of  engineering,  and  the  proper  amount  to  allow  for 
depreciation  is  largely  a  matter  of  guesswork  based  upon  previous 
experience.  A  sinking  fund  for  the  creation  of  a  depreciation 
reserve  should  be  formed  by  placing  annually  at  compound  in- 
terest a  certain  sum  of  money  which,  at  the  end  of  the  estimated 
life  of  the  structure  or  plant,  will  reproduce  the  sum  originally 
invested.  The  accompanying  table  has  been  worked  out  on  a 
basis  of  5  per  cent,  compound  interest.  It  gives  the  amount,  in 
dollars,  which  must  be  put  aside  each  year  in  order  to  provide  a 
fund  of  $100  at  the  end  of  a  term  of  years  after  which  the  value  of 
the  works  or  materials  under  consideration  is  assumed  to  be  nil. 


62  ELECTRIC  POWER  TRANSMISSION 

DEPRECIATION  TABLE 

(On  basis  of  5  per  cent,  compound  interest  earned  by  money  put   aside 
annually) 


Life,  years 

Depreciation, 
per  cent. 

Life,  years 

Depreciation, 
per  cent. 

2 

48  70 

28. 

1  710 

4 

23  20 

30  

1  505 

6  
8  

14.70 
10.50 

32  
34  

1.325 
1.175 

10 

7  95 

36 

1  045 

12 

6  28 

38 

0  928 

14  

5.10 

40  

0.828 

16  
18  
20  
22 

4.23 
3.55 
3.03 
2  60 

42  
44  
46  
48 

0.740 
0.662 
0.593 
0  532 

24 

2  25 

50 

0  477 

26 

1  96 

Although  it  is  rarely  necessary  to  consider  scrap  values,  an 
exception  should  be  made  in  the  case  of  the  copper  conductors  of 
transmission  lines.  If  D  is  the  price  originally  paid  for  the 
material,  and  S  is  the  estimated  scrap  value  at  the  end  of  n 
years,  the  percentage  of  the  original  sum  D  to  be  put  aside 
annually  to  cover  depreciation  is  not  r  per  cent. — as  calculated 
for  zero  value  at  the  end  of  n  years — but  rf  per  cent,  of  which  the 
value  is 


The  "life,"  in  years,  of  any  part  of  a  machine  or  structure  is 
very  difficult  to  estimate.  It  is  here  that  the  distinction  between 
natural  and  functional  depreciation  becomes  important,  because 
whichever  one  appears  to  indicate  the  shortest  life,  should  be 
considered  to  the  exclusion  of  the  other.  Thus,  the  life  of  wood 
poles — liable  to  decay  and  attacks  by  insects — will  lead  to  the 
allowance  for  natural  depreciation  being  larger  than  for  func- 
tional depreciation,  and  the  latter  can  therefore  be  ignored.  But 
there  are  many  kinds  of  plants,  such  as  generators  of  inefficient 
design  or  insufficient  capacity,  of  which  the  life  determined  on  a 
basis  of  functional  depreciation  is  shorter  than  their  probable 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    63 

wearing  possibilities,  and  it  is  then  the  natural  depreciation  which 
should  be  ignored. 

33.  Example:  Method  of  Determining  Most  Economical 
Voltage. — Consider  the  case  of  a  typical  medium-head  hydro- 
electric power  station: 

Distance  of  transmission  =  50  miles. 

Duplicate  three-phase  line  with  copper  conductors. 

Cost  of  copper  conductors  =  $20  per  100  Ib. 

Power  demanded  =  15,000  hp.  or  11,200  k.w.  (It  is  assumed 
that  this  power  will  be  required  continuously  day  and  night  for 
industrial  purposes,  and  that  it  is  the  probable  limit  of  the  water- 
power  available.) 

Power  factor  =  0.8. 

Selling  price  of  power  =  $21  per  horsepower-year. 

Interest  on  capital  invested;  allow  6  per  cent. 

The  economic  drop  of  voltage  per  mile  of  single  conductor  as 
given  by  formula  (19)  is: 


Er  =  g.35J<L><P 

\        Pi 

Where  p  is  the  price  in  dollars  of  100  Ib.  weight  of  conductor  (in 
this  example  p  =  20),  a  is  the  percentage  to  cover  annual  depre- 
ciation and  interest  on  cost  of  conductors,  and  p\  is  the  cost  per 
kilowatt-year  of  the  wasted  power.  The  proper  value  for  a 
may  be  arrived  at  by  estimating  the  term  of  years  corresponding 
to  the  life  of  the  conductors,  at  the  end  of  which  they  are  supposed 
to  be  of  no  value.  Taking  16  years  as  the  life  of  the  conductors, 
the  depreciation  to  be  allowed  according  to  the  table  is  4.23 
which  would  be  the  proper  value  to  take  on  the  basis  of  5  per 
cent,  compound  interest  if  the  copper  wire  has  no  scrap  value  at 
the  end  of  this  time.  It  is  very  difficult  to  estimate  the  scrap  value 
of  conductors  16  years  ahead.  Apart  from  the  market  quotations 
which  may  then  determine  the  price  per  pound  of  the  metal,  the 
fact  that  the  transmission  line  is  very  likely  a  long  way  from  the 
place  where  there  is  a  demand  for  the  copper  must  not  be  over- 
looked. Not  only  must  the  labor  cost  of  removing  the  wires  from 
the  poles,  together  with  the  transportation  charges,  be  deducted 
from  the  price  obtainable  for  the  copper,  but  a  further  deduction 
should  usually  be  made  to  cover,  in  whole  or  in  part,  the  cost 
of  stringing  the  new  conductors.  Assuming  the  net  amount 
likely  to  be  obtained  from  the  sale  of  the  scrap  copper  to  be  $6 


64  ELECTRIC  POWER  TRANSMISSION 

per    100    Ib.    the    proper    allowance   for   depreciation    will    be 
4.23  X  14 
-20— 
which  makes 

a  =  6  +  2.96  =  (say)  9  per  cent. 

With  regard  to  pi.  if  the  demand  for  power  were  equal  to  the 
available  supply  from  the  time  of  the  power-plant  being  put  into 
operation,  the  works  cost  of  waste  power  would  be  the  same  as 
the  selling  price;  but,  if  we  assume  that  the  supply  exceeds  the 
demand  during  the  first  four  years  of  operation,  and  that  the 
cost  of  waste  power  during  this  period  is  only  $7  per  horsepower- 
year,1  the  average  cost  of  wasted  power  during  the  16  years  life  of 
the  conductors  should  be  arrived  at  by  estimating  the  current  and 
power  loss  for  each  year  that  the  plant  is  in  operation. 

A  first  approximation  to  the  required  line  voltage  may  be 
obtained  by  formula  (23) : 

/,.  ,  ,    kilowatts 

Kilovolts  =  5.5  ^/distance  + 


100 


=  70 
The  current  will  be, 

kilowatts  transmitted 


I  = 


V3  X  70  X  0.8 
kilowatts 


97 

The  line  losses  will  be  proportional  to  /2,  and  in  order  to  arrive 
at  a  suitable  value  for  pi  for  use  in  formula  (19),  the  demand  for 
power  during  the  first  four  years  of  operation  (before  the  hy- 
draulic plant  is  utilized  to  its  possible  limit)  should  be  esti- 
mated, and  a  table  constructed  as  below.  An  average  figure 
may  be  assumed  for  power  supplied  during  any  given  period  of 
twelve  months. 

1  The  actual  works  cost  of  the  wasted  power  is  always  difficult  to  deter- 
mine exactly.  It  must,  however,  be  remembered  that  even  with  unlimited 
power,  and  no  appreciable  increase  in  maintenance  and  operating  charges 
with  increase  of  losses,  the  greater  capital  cost  of  the  plant  installed  to 
provide  this  waste  power  has  to  be  taken  into  account  and  expressed  in  the 
form  of  an  annual  charge  per  kilowatt  wasted,  whether  this  waste  occurs 
in  the  generating  and  transforming  plant  or  the  line  itself. 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    65 


1 

2 

3 

4 

Period 

Estimated 
kilowatts 
demanded 

Current  7 

I-  X  years 

1st  year  
2dyear  
3d  year 

4,000 

5,000 
6  000 

41.3 
51.6 
61  8 

1,710 
2,660 
3  820 

4th  year 

8  000 

82  5 

6  800 

5th  to  16th  year  

11,200 

115.5 

160,000 

Total.  =  174,990  or,  say,  175,000 

The  total  of  the  figures  in  the  last  column  covering  the  four 
years  during  which  the  cost  of  waste  power  is  estimated  at  $7 
per  horsepower-year,  is  14,990,  or  say  15,000,  as  compared  with 
160,000  for  the  period  of  12  years  during  which  the  cost  of  the 
wasted  power  will  be  $21  per  horsepower-year.  A  reasonable 
value  to  take  for  pi  is,  therefore, 


(15  X  7)  +  (160  X  21) 
175  X  0.746 


=  $19.8 


where  the  figure  0.746  is  merely  for  the  purpose  of  converting 
cost  per  horsepower  into  cost  per  kilowatt. 

The   economic  resistance  pressure  drop,  by  formula  (19)  is 
therefore 


=  28.2  volts  per  mile 

It  is  well  to  note  that  the  economic  voltage  drop  does  not 
correspond,  in  this  particular  example,  to  the  full  load  ohmic 
drop  of  pressure.  The  current  which  causes  the  ohmic  drop 
of  28.2  volts  per  mile  may  be  calculated  as  follows.  The  average 
value  of  72  is  the  total  of  column  4  in  the  above  table,  divided 


by  the  number  of  years,  namely, 


175,000 
16 


=  10,930,  the  square 


root  of  which  is  104.5,  and  this  is  the  figure  for  current  to  be  used 
in  the  preliminary  power-loss  calculations,  instead  of  115.5  which 
is  the  full  load  current.  The  line  may  therefore  be  considered 
as  transmitting  continuously  \/3  X  70  X  0.8  X  104.5  =  10,130 
or,  say,  10,000  kilowatts. 


66  ELECTRIC  POWER  TRANSMISSION 

When  the  section  of  the  conductors  is  such  as  to  satisfy  Kel- 
vin's law  of  economy,  the  yearly  cost  of  the  PR  losses  is  equal 
to  the  amount  representing  annual  depreciation  and  interest 
on  first  cost  of  conductors;  and  the  total  annual  charges  on  active 
line  material,  for  a  three-phase  line,  will  therefore  be: 

Pi  X  PRL 
1000 

where  R  is  the  resistance  per  mile  of  conductor.     But 

/  =          P  X  1000 
~  A/3  X  E  X  cos  0 

where  P  stands  for  the  kilowatts  transmitted. 

Also  :  IR  =  voltage  loss  per  mile  =  ET.  So  that  the  formula 
for  the  total  yearly  charges  on  conductors  may  be  written 


.     . 


E  X  cos  6 
which  in  this  example  becomes 

2  X  V3  X  28.2  X  10,000  X  19.8  X  50  _ 
70,000  X  0.8 

An  amount  which  is  independent  of  the  fact  that  the  trans- 
mission, in  this  particular  instance,  is  by  two  three-phase  lines 
ordinarily  connected  in  parallel. 

34.  Closer  Estimate  of  Economical  Voltage. — In  order  to  take 
into  account  first  cost,  life,  annual  maintenance,  and  operating 
charges  of  every  portion  of  the  complete  undertaking  which  may 
be  affected  by  a  change  in  the  transmission  voltage,  the  costs, 
worked  out  on  an  annual  basis,  may  be  arranged  in  tabular 
form  as  here  shown,  where  the  total  charges  for  the  70,000- 
volt  scheme  are  compared  with  the  estimated  charges  for  an 
88,000-volt  transmission.  In  this  particular  example,  the  figures 
are  favorable  to  the  lower  voltage;  but  the  difference  is  very 
small. 

By  increasing  the  voltage  of  transmission  from  70,000  to  88,000 
volts,  a  saving  of  $3690  is  effected  on  the  annual  charges  if  the 
copper  conductors  alone  are  considered;  but  the  increased  cost 
of  other  portions  of  the  complete  plant,  due  to  the  raising  of  the 
line  pressure,  results  in  an  actual  increase  of  the  total  annual 
charges  thus  showing  the  pressure  first  chosen  to  be  preferable 


ECONOMIC  PRINCIPLES  AND  CALCULATIONS    67 


68  ELECTRIC  POWER  TRANSMISSION 

to  a  higher  pressure.  The  process  could  be  repeated  for  a 
voltage  lower  than  70,000;  but  the  very  small  difference  in  favor 
of  this  pressure  as  compared  with  88,000  volts  indicates  that 
any  further  appreciable  reduction  of  the  transmission  voltage 
would  almost  certainly  lead  to  a  higher  annual  cost. 

It  will  be  understood  that  the  accompanying  estimate  of  total 
annual  charges  of  the  two  selected  voltages  does  not  include  any 
items  other  than  those  that  are  liable  to  vary  with  changes  in 
the  line  voltage.  An  estimate  covering  the  complete  under- 
taking would,  in  addition  to  the  items  named,  have  to  take 
account  of  riparian  rights  for  dam,  reservoir,  etc.,  preliminary 
legal  and  other  expenses;  cost  of  providing  proper  access  for  ma- 
terials to  site  of  works;  dam  and  hydraulic  works  outside  station 
building;  turbines;  electric  generators  and  exciters;  auxiliary 
plant;  sundries  and  contingencies. 

In  the  case  of  a  short  distance  transmission  with  a  line  pressure 
not  exceeding  11,000  volts,  and  the  possibility  of  winding  the 
generators  for  the  full  pressure,  the  relative  costs  and  efficiencies 
of  generators  wound  for  different  voltages  should  be  taken  into 
account. 


CHAPTER  IV 
ELECTRICAL  PRINCIPLES  AND  CALCULATIONS 

35.  Materials. — Under  ordinary  circumstances,  the  choice  of 
material  for  the  conductors  of  an  overhead  H.T.  transmission 
line  lies  between  copper  and  aluminum.  Under  certain  condi- 
tions, as  for  the  transmission  of  continuous  currents  or  when  the 
price  of  more  suitable  materials  is  abnormally  high,  galvanized 
iron  or  steel  may  prove  satisfactory  and  economical;  and  com- 
pound wires  or  cables  such  as  copper-clad  steel,  and  aluminum 
cables  with  galvanized  steel  core,  are  used  where  great  mechanical 
strength  is  of  more  importance  than  high  conductivity.  Much 
has  been  written  on  the  relative  advantages  of  copper  and 
aluminum  for  transmission-line  conductors,  and  some  writers, 
who  have  not  been  interested  in  the  sale  or  manufacture  of  con- 
ductor materials,  have  no  doubt  treated  the  subject  impartially, 
and  stated  the  case  for  either  metal  with  clearness  and  ability; 
but  there  is  usually  a  tendency  to  give  too  much  weight  to  the 
question  of  first  cost.  It  is  very  difficult  to  make  a  comparison 
which  shall  be  of  general  utility,  between  various  metals,  because 
not  only  the  electrical,  but  also  the  mechanical,  properties  have 
to  be  taken  into  account,  and  the  requirements  in  the  latter 
respect  will  depend  largely  on  local  conditions.  Then  again, 
with  market  fluctuations,  and  tariffs  controlling  the  prices  of 
raw  materials  in  different  countries,  together  with  varying  costs 
of  freight  from  manufacturers'  works,  a  comparison  of  costs, 
except  when  based  on  current  quotations,  is  of  little  value.  For 
these  reasons  no  direct  comparison  between  conductors  of  differ- 
ent materials  will  be  made  here,  but  leading  particulars  will 
be  given,  together  with  such  notes  as  the  writer's  experience 
may  suggest,  which  it  is  hoped  will  be  helpful  to  the  transmission- 
line  engineer  in  deciding  upon  the  right  material  to  use  under 
given  circumstances.  Tables  of  resistances,  sizes  and  weights, 
and  other  physical  properties  of  the  materials  will  be  found  in  the 
various  engineering  handbooks  and  manufacturers'  catalogues; 
and  only  such  particulars  will  be  given  here  as  may  be  useful 
for  preliminary  calculations. 

69 


70 


ELECTRIC  POWER  TRANSMISSION 


36.  Copper. — It  is  probably  safe  to  assert  that,  apart  from 
the  question  of  cost,  the  high  conductivity  combined  with  the 
great  strength  and  elasticity  of  hard-drawn  copper,  give  this 
material  the  advantage  over  all  others  for  use  on  the  average 
high-tension  electric  transmission  line. 

The  ultimate  tensile  strength  of  hard-drawn  copper  is  greater 
per  square  inch  of  section  in  the  smaller  wires,  being  approxi- 
mately as  follows: 


Gauge  No.,  B.  &  S. 

Diameter,  inches 

Breaking  stress, 
Ib.  per  sq.  in. 

000 

0.410 

52,000 

0 

0.325 

55,000 

2 

0.258 

58,000 

4 

0.204 

60,000 

6 

0.162 

62,000 

8 

0.128 

64,000 

10 

0.102 

65,000 

12 

0.081 

66,000 

14 

0.064 

67,000 

16 

0.051 

67,500 

18 

0.040 

68,000 

A  stranded  cable,  in  which  the  pitch  is  usually  between  12  and 
16  diameters  of  the  cable,  will  generally  break  under  a  load 
slightly  smaller  than  the  combined  breaking  loads  of  the  in- 
dividual wires.  The  tensile  strength  of  a  stranded  cable  should, 
however,  not  be  less  than  90  per  cent,  of  the  combined  strengths 
of  the  single  wires. 

The  elastic  limit  of  hard-drawn  copper  wires  is  about  60 
per  cent,  of  the  breaking  stress;  but  it  may  be  as  high  as  70  per 
cent,  and  even  75  per  cent,  of  the  ultimate  stress. 

37.  Aluminum.1 — The  conductivity  of  hard-drawn  aluminum 
wire  is  between  60  per  cent,  and  61^  per  cent,  by  Matthiessen's 
standard;  pure  copper  being  100  per  cent.  The  weight  of  an 
aluminum  conductor  is  almost  exactly  half  that  of  the  copper 
conductor  of  equal  resistance,  and  it  is  about  77  per  cent,  as 
strong  as  the  equivalent  copper  cable  (safe  working  stress). 

1  Valuable  information  regarding  the  properties  and  uses  of  Aluminum 
wire  will  be  found  in  the  publication  entitled  "From  the  Falls  to  the  Factory," 
issued  by  the  British  Aluminium  Company,  Ltd.  of  London,  England,  and 
Toronto,  Canada. 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS   71 

Comparing  aluminum  of  61  per  cent,  conductivity  with  copper 
of  97  per  cent,  conductivity,  the  diameter  of  the  equivalent 
aluminum  cable  would  be  1.26  times  the  diameter  of  the  copper 
cable. 

The  ultimate  tensile  stress  of  hard-drawn  aluminum  wire 
usually  lies  between  24,000  and  32,000  Ib.  per  sq.  in.,  de- 
pending upon  the  size  of  wire  and  hardness;  if  carried  beyond 
a  certain  point,  high  tensile  strength  is  a  disadvantage,  because 
the  conductivity  is  lowered  and  the  wire  becomes  "short." 
Some  recent  tests  made  on  the  strands  composing  an  aluminum 
cable  of  61  per  cent,  conductivity  gave  the  following  results: 

Diameter  of  wire  (inch) 0.1092  0. 116  0. 138 

Number  of  tests 8       33 

Breaking  stress,  highest 34,500       28,900 

Breaking  stress,  lowest 28,200       24,200 

Breaking  stress,  average 32,100  29,300  26,100 


The  elastic  limit  of  hard-drawn  aluminum  wire  is  from  50  to 
60  per  cent,  of  the  breaking  stress. 

Aluminum  is  readily  attacked  by  alkaline  substances,  and 
coils  of  cable  should  not  be  left  lying  on  wet  marshy  ground  liable 
to  contain  alkalies,  or  in  old  stables  where  ammonia  may  be 
present. 

Aluminum  is  not  easily  soldered,  because  of  the  thin  film  of 
oxide  which  quickly  forms  on  the  surface  exposed  to  the  atmos- 
phere. The  tin  must  be  mechanically  worked  through  the  oxide 
coating  with  the  aid  of  an  old  file  or,  preferably,  a  scratch  brush 
with  bristles  of  0.01  in.  diameter  steel  not  more  than  1  in.  long. 
A  little  experience  is  needed  for  neatly  soldering  aluminum  into 
cable  sockets,  etc.,  partly  for  the  above  reason  and  also  because 
the  metal  is  a  good  conductor  of  heat,  and  the  parts  to  be  tinned 
will  cool  rapidly  unless  special  precautions  are  taken. 

38.  Iron  and  Steel. — The  ordinary  commercial  galvanized 
steel  strand  cable,  as  used  for  guy  wires,  has  a  breaking  strength 
averaging  70,000  Ib.  per  sq.  in.,  and  a  conductivity  of  about 
ll/^  per  cent,  by  Matthiessen's  scale.  When  used  to  con- 
vey alternating  currents,  the  high  permeability  of  iron  in- 
creases the  so-called  skin  effect,  with  the  result  that  the  resistance 
to  the  flow  of  current  may  be  greatly  increased,  depending  upon 
the  size  of  the  cable  and  the  frequency.  Apart  from  the  greater 


72  ELECTRIC  POWER  TRANSMISSION 

loss  of  voltage  due  to  apparent  increase  of  resistance  when  iron 
wires  are  used  with  alternating  currents,  the  loss  of  pressure  due 
to  increased  reactance  must  also  be  taken  into  account.  The 
external  reactance  is  the  same  for  a  given  diameter  and  spacing 
of  wires  whatever  may  be  the  material;  but  the  internal  reactance 
will  obviously  be  much  greater  for  a  "magnetic"  than  for  a 
"non-magnetic"  conductor.  This  point  will  be  taken  up  again 
in  a  later  article.  When  comparing  iron  with  copper  or  alu- 
minum as  a  possible  material  for  conductors,  the  shorter  "life" 
of  the  iron  wire  must  not  be  overlooked.  A  good  quality  of 
galvanized  wire  or  stranded  conductor  should  be  used,  such  as  the 
E.  B.  B.  (Extra  Best  Best)  grade  of  which  particulars  are  given 
in  the  following  tables. 

The  weight  of  an  iron  or  steel  cable  will  be  at  least  5  times 
that  of  the  copper  cable  of  equal  resistance,  and  with  the  higher 
grade  (and  stronger)  steels,  this  multiplier  may  be  as  high  as  10. 
High-grade  steel  conductors  can  be  used  to  advantage  for  very 
long  spans,  or  where  the  climatic  conditions  are  such  as  to 
subject  the  cables  to  abnormally  great  stresses.  Extra  high 
strength  steel  wire  can  be  obtained  with  an  ultimate  strength  of 
180,000  Ib.  per  sq.  in.  and  an  elastic  limit  of  110,000  Ib.  per 
sq.  in.  A  possible  maximum  working  stress  for  this  material 
would  be  about  80,000  Ib.  per  sq.  in. 

Whatever  may  be  the  material  of  the  conductor,  a  stranded 
cable  made  up  of  a  large  number  of  small  wires  will  be  stronger 
than  a  cable  of  the  same  sectional  area  made  up  of  fewer  large 
wires. 

39.  Copper-clad  Steel. — By  welding  a  coating  of  copper  on  a 
steel  wire,  a  compound  wire  known  as  hard-drawn  copper-clad 
steel  wire  is  produced.  This  has  been  well  tested,  and  ex- 
perience has  shown  it  to  be  an  excellent  material  for  many 
purposes.  The  wire  can  be  made  up  in  the  form  of  cables  if 
desired,  which,  when  used  as  conductors  for  overhead  transmis- 
sions, will  have  greater*  strength  than  cables  made  entirely  of  cop- 
per, and  lower  resistance  than  cables  made  entirely  of  steel.  The 
two  metals  are  intimately  and  permanently  welded  together 
by  means  of  a  special  copper-iron  alloy,  and  the  relative  quanti- 
ties so  adjusted  that  the  finished  wire  has  a  conductivity  of  30 
per  cent,  to  40  per  cent,  of  a  copper  wire  of  the  same  diameter. 
The  ultimate  tensile  strength  of  commercial  copper-clad  wire  of 
various  sizes  is  approximately  as  below : 


ELECTRICAL  PRINCIPLES  AND  CALC ULA  TIONS   73 


Gauge  No.,  B.  &  S. 

Diameter,  in. 

Breaking  weight,  Ib. 

stronger  than  copper 
of  same  o!ia  meter 

000 

0.410 

7600 

1.15 

0 

0.325 

5400 

1.20 

2 

0.258 

3700 

1.23 

4 

0.204 

27001 

6 

0.162 

1750  / 

1.38 

8 

0.128 

12001 

10 

0.102 

780  / 

1.47 

40.  Stranded  Cables  with  Steel  Wire  Core. — The  central  wire 
of  a  stranded  conductor  may  be  galvanized  steel,  or  a  small 
diameter  steel  cable  may  be  used  for  the  core.  This  increases 
the  strength,  especially  in  the  case  of  aluminum  cables,  and  a 
compound  conductor  of  this  sort  is  useful  for  long  spans  on  an 
aluminum  wire  transmission  line. 

It  is  usual  to  neglect  the  current-carrying  capacity  of  the  steel 
core,  and  calculate  the  conductivity  on  the  assumption  that  all 
the  current  is  carried  by  the  strands  of  the  higher  conductivity 
metal.  Composite  cables  can  be  made  of  steel  and  copper  wires, 
but  the  strength  of  hard-drawn  copper  is  so  great  that  the  gain 
due  to  the  addition  of  the  steel  core  is  comparatively  small. 

The  impedance  of  a  steel  core  conductor  will  be  higher  than 
that  of  a  conductor  made  entirely  of  non-magnetic  material,  but 
experiment  has  shown  that  the  increase  of  impedance  is  prac- 
tically negligible  when  there  are  two  layers  of  copper  wires 
spiralled  in  reverse  directions  on  a  central  steel  core;  it  would 
seem  as  if  the  current  divided  itself  in  two  equal  parts  circulating 
in  opposite  directions,  thus  neutralizing  any  tendency  to  magnet- 
ize the  steel  core. 

Hemp  Core  Cables. — When  a  stranded  conductor  is  made  up 
of  one  material  only,  the  central  wire  is  subjected  to  a  greater 
strain  than  the  wires  that  are  spiralled  around  it.  This  difficulty 
can  be  overcome  by  using  hemp  for  the  central  core.  A  hemp 
core  cable  will  have  slightly  increased  diameter  for  the  same 
conductivity  and  greater  smoothness  of  surface  than  a  metal 
core  cable,  and  these  features  will  raise  the  critical  voltage  at 
which  corona  will  form. 

For  particulars  relating  to  underground  cables,  the  reader  is 
.  referred  to  Chapter  VII. 


74  ELECTRIC  POWER  TRANSMISSION 

41.  Physical  Constants  and  Sizes  of  Commercial  Conductors.— 

The  accompanying  table  gives  the  most  important  physical 
constants  for  various  conductor  materials.  It  will  be  noticed 
that  aluminum  has  a  larger  temperature  coefficient  than  copper. 
This  has  an  important  bearing  on  the  economic  length  of  span ; 
the  difference  in  sag  between  sutaimer  and  winter  temperatures 
is  often  considerable  with  aluminum  conductors,  but  this  dif- 
ference is,  of  course,  more  noticeable  on  the  shorter  spans  such 
as  occur  with  a  wood  pole  construction:  on  long  spans,  the 
difference  in  sag  due  to  temperature  changes  is  very  small,  what- 
ever metal  is  used. 

An  argument  often  advanced  in  favor  of  aluminum  conductors 
is  that  the  weight  of  these,  for  any  given  transmission  scheme, 
is  only  about  half  that  of  copper.  This  is  certainly  an  advantage 
in  the  handling  of  the  wire,  but  otherwise  it  is  at  least  counter- 
balanced by  the  fact  that  the  wind  effect  is  greater  on  the  in- 
creased diameter  and  that  the  towers  must  often  be  higher  than 
if  copper  is  used,  partly  on  account  of  the  higher  coefficient  of 
expansion  of  aluminum,  but  mainly  because  of  the  lower  per- 
missible stress.  The  advantage  of  lighter  weight  is  largely  dis- 
counted by  the  fact  that  the  equivalent  aluminum  conductor 
can  only  be  drawn  up  to  a  tension  equal  to  about  three-quarters 
of  the  permissible  maximum  tension  of  the  copper  cable.  On 
the  other  hand,  the  larger  diameter  of  the  aluminum  cable  may 
be  an  advantage  on  very  high-pressure  transmissions,  because 
it  raises  the  critical  voltage  at  which  corona  losses  become 
appreciable. 

It  has  been  customary  in  the  United  States  for  those  in  control 
of  the  metal  markets  to  regulate  the  price  of  aluminum  so  that 
there  shall  be  no  economic  advantage  in  using  it  to  replace 
copper;  but  it  is  used  by  several  power  companies,  among  which 
may  be  mentioned  the  Pacific  Light  and  Power  Co.  and  the  South- 
ern Sierras  Power  Go.  On  the  continent  of  Europe  and  in 
Canada  aluminum  has  found  favor  and  is  much  used. 

The  accompanying  wire  table  gives  the  approximate  resist- 
ances and  weights  of  the  usual  sizes  of  cable,  whether  of  copper  or 
aluminum,  but  makers'  lists  should  be  consulted  for  exact  par- 
ticulars, as  the  method  of  building  up  the  stranded  conductors 
necessarily  modifies  to  a  small  extent  the  average  figures  here 
given.  The  figures  in  the  table  are  intended  for  quick  slide  rule 
calculations,  and  the  resistances  are  approximately  correct  for 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS   75 


S  s 


ill 


111 


11! 
Ill 

,!•&& 


o  2~~ 
x  V'M "" 
3 


CO   <N    0   T* 
«  O  O  0 


'8* 

CD   O 


o       o££ 
X      X£2 


S"  .2  o"  o' 
o  oo 


o  2=2  a 

x  x$2S 

Q  t^»  C^  O  ^ 

N  CO   C5   O   O 


O    o  O   O  ' 


(NO-* 

bod 


51  ^  O  fi  69  ^        t>^ 

d  CO    (N     ^H     ^H  ^H 


°-  °-  °-  -2  °-      v      v  w  o  ic      ^ 

tf"  >*  SB  Q          "3          ^          ^(Ni-HGO          ^OO 

o»OC<>CO  CO  "3  COCOON  OOO 


*     ''•  I 
3     :  «   e 


.15  •- 


1     :^x.2^ 

jii-li!- 


II 


ii 


I    ill^Ml  irl 

•5  •§  -5  w  ^>  "S  e  J  ,  •   >   > 


lil^ll 

.2r  TS  -d  „     13    sS 


I 

1  I 


I  I 
1 


o       .9 

II    | 

fl    s^         «5 

Si     -3 

S-S      | 

s^  .2 
S28& 

5     «    -^      rti 


ill 

i|  $ 

•I  I » 

s   -S.g 

i«l 


76 


ELECTRIC  POWER  TRANSMISSION 


a  temperature  of  60°  F.  The  sizes  of  the  smaller  conductors  are 
given  in  the  B.  &  S.  gauge  because  this  is  generally  used  on  this 
continent.  With  this  system,  when  the  area  of  any  particular 
gauge  number  is  known,  it  is  only  necessary  to  double  this  in  order 
to  get  the  area  of  the  third  size  larger;  or  if  instead  of  multi- 
plying by  two,  the  multiplier  1.261  is  used,  this  will  give  the 
area  of  the  next  size  larger  in  the  B.  &  S.  series.  It  is  convenient 
to  remember  that  No.  10  B.  &  S.  copper  wire  measures  almost 
exactly  ^f  o  m-  m  diameter,  and  has  a  resistance  of  1  ohm  per 
1000  ft. 

When  the  resistance,  R,  per  mile  of  a  stranded  conductor  is 
known,  the  weight  per  mile  is  approximately: 


For  Copper;  pounds  per  mile=  -5- 


440 


For  Aluminum;  pounds  per  mile  =  ~rr 


RESISTANCE  AND  WEIGHT  OP  STRANDED  CONDUCTORS 


Size,  cir.  mils 
and  B.  &  S. 
gauge 

Diameter, 
inches, 
approx. 

Circular 
mils, 
nominal 

Area, 
sq.  in., 
approx. 

Copper 

Aluminum 

Ohms 
per 
mile 

Weight 
per 
m,le, 

Ohms 

3L 

Weight 
per 
mile, 
Ib. 

600,000 

0.89 

600,000 

0.472 

0.0920 

9750 

0.153    2920 

500,000 

0.81 

500,000 

0.393 

0.1095 

8100 

0.182     2430 

450,000 

0.77 

450,000 

0.354 

0.1210 

7300 

0.202     2187 

400,000 

0.73 

400,000 

0.314 

0.1363 

6500 

0.227 

1944 

350,000 

0.68 

350,000 

0.275 

0.1566 

5650 

0.260 

1701 

300,000 

0.63 

300,000 

0.236 

0.1818 

4880 

0.303 

1458 

250,000 

0.58 

250,000 

0.1965 

0.2192 

4060 

0.364 

1215 

4/0 

0.53 

211,600 

0.1661 

0.260 

3448 

0.430 

1028 

3/0 

0.47 

167,800 

0.1317 

0.326 

2730 

0.542 

816 

2/0 

0.42 

133,100 

0.1045 

0.410 

2165 

0.684 

647 

0 

0.37 

105,600 

0.0830 

0.518 

1705 

0.862 

513 

1 

0.33 

83,700 

0.0657 

0.655 

1346 

1.085 

407 

2 

0.29 

66,400 

0.0521 

0.826 

1067 

1.370 

323 

3 

0.26 

52,600 

0.0413 

1.040 

850 

1.728 

256 

4 

0.23 

41,700 

0.0327 

1.313 

675 

2.185 

203 

5 

0.207 

33,090 

0.0260 

1.635 

540 

2.720 

162 

6 

0.183 

26,250 

0.0206 

2.091 

422 

3.470 

126 

ELECTRICAL  PRINCIPLES  AND  CALC  ULA  TIONS   77 

42.  Skin  Effect.  —  Imagine  a  straight  length  of  cable  of  fairly 
large  cross-section,  through  which  a  steady  continuous  current 
is  flowing,  the  return  circuit  being  a  considerable  distance  away. 
The  magnetic  induction  due  to  this  current  will  not  be  only  in 
the  non-conducting  medium  surrounding  the  wire,  but  a  certain 
amount  —  due  to  the  current  in  the  central  portions  of  the  cable  — 
will  be  in  the  substance  of  the  conductor  itself.  In  other  words, 
the  magnetic  flux  surrounding  one  of  the  central  strands  of  the 
cable  will  be  greater  than  that  which  surrounds  a  strand  of  equal 
length  situated  near  the  surface.  It  follows  that,  if  the  circuit 
be  now  broken,  the  current  will  die  away  more  quickly  near  the 
surface  of  the  conductor  than  at  the  center;  and,  for  the  same 
reason,  on  again  closing  the  circuit,  the  current  will  spread  from 
the  surface  inward. 

If,  now,  the  conductor  be  supposed  to  convey  an  alternating 
current,  it  is  evident  that,  with  a  sufficiently  high  frequency 
(or  even  with  a  low  frequency  if  the  conductor  be  of  large  cross- 
section),  the  current  will  not  have  time  to  penetrate  to  the  in- 
terior, but  will  reside  chiefly  near  the  surface.  This  crowding 
of  the  current  toward  the  outside  portions  of  the  conductor 
has  the  effect  of  apparently  increasing  the  resistance;  and  it 
follows  that  if  I  is  the  total  current  in  a  cable  of  ohmic  resistance 
R,  the  power  lost  in  watts  would  no  longer  be  I2R,  as  in  the  case  of 
a  steady  current,  but  IZR',  where  R'  —  which  stands  for  the 
apparent  resistance  of  the  conductor  —  is  k  times  greater  than 
R,  its  true  resistance.  The  multiplier  k  may  be  read  off  the 
diagram  Fig.  20,  or  if  preferred,  it  can  be  calculated  by  means  of 
the  formula: 


(24) 


where  F  is  a  factor  proportional  to  the  vertical  distances  on 
the  diagram,  that  is  to  say,  to  the  quantity  area  of  cross-section 
X  frequency.  The  value  of  F  for  copper  is  : 

F  =  0.0105d2/ 
and  for  aluminum, 

F  =  0.0063d2/ 

where  d  is  the  diameter  of  the  conductor  in  inches,  and  /  is 
the  frequency  in  periods  per  second.  This  formula  and  the 
curves  of  Fig.  20  are  based  on  the  assumption  that  the  return 
current  is  at  an  infinite  distance;  but  this  assumption  introduces 


78 


ELECTRIC  POWER  TRANSMISSION 


no  appreciable  error  when  dealing  with  overhead  transmission 
lines. 

It  will  be  observed  that,  so  long  as  the  product  dzf  remains 
unaltered,  the  multiplier  k  is  constant  provided  the  material 


!• 


1.00     L02    1.04     1.06    1.08    1.10    1.12    1.14     1.16     1.18     1.20     1.22    1.24     1.26    L28     1.30 
"Skin  Effect"  Multiplier  (&) 

FIG.  20. — Diagram  giving  "skin  effect"  coefficient. 

remains  the  same.  Thus  if,  when  doubling  the  frequency, 
the  sectional  area  of  the  (circular)  conductor  is  halved,  the 
resistance  to  alternating  currents 


ratio 


resistance  to  continuous  currents 


remains  unaltered. 


ELECTRICAL  PRINCIPLES  AND  CALC  ULA  TIONS   79 

In  regard  to  the  material  of  the  conductor,  the  value  of  F 
in  the  formula  is  directly  proportional  to  the  specific  conduc- 
tivity of  the  metal  so  long  as  the  frequency  remains  constant. 
Thus  if  F  (or  the  value  of  the  ordinates  in  the  diagram,  Fig.  20) 
is  known  for  a  conductor  of  given  diameter,  made  of  copper,  its 
value  for  any  other  "non-magnetic"  material  is  given  by  the 
ratio: 

conductivity  of  metal  of  conductor 
conductivity  of  copper 

If  the  conductor  is  of  iron  (or  other  "magnetic"  material), 
the  value  of  k  may  be  much  greater  than  this  ratio  would  indicate. 
This  point  will  be  taken  up  again  in  Article  44. 

It  is  a  not  uncommon  belief  that  when  aluminum  conductors 
are  used  in  place  of  copper,  the  larger  diameter  necessary  to 
give  the  same  conductivity  will  lead  to  a  greater  loss  through 
"skin  effect;"  but  the  above  multiplying  ratio  makes  it  clear 
that  the  percentage  increase  of  losses  with  alternating  currents 
of  the  same  frequency  will  be  independent  of  the  material  of 
the  conductor  (iron  excepted),  because  the  greater  sectional 
area  necessary  to  maintain  the  same  ohmic  resistance  of  the 
lines  when  a  wire  of  lower  conductivity  is  used,  is  evidently 
exactly  balanced  by  the  higher  specific  resistance  of  the 
metal. 

The  increased  pressure  drop  and  PR  loss  on  overhead  lines 
at  normal  frequencies  and  with  conductors  of  average  size 
are  usually  very  little  greater  with  alternating  than  with  con- 
tinuous currents;  but  when  the  material  is  iron  or  steel  the 
difference  may  be  very  noticeable,  and  in  such  cases  as  the 
rail  return  of  an  alternating  current  traction  system,  it  should 
be  taken  into  account. 

43.  Inductance  of  Transmission  Lines. — For  the  purpose  of 
calculating  the  flux  of  induction  outside  a  straight  cylindrical 
conductor,  it  is  permissible  to  assume  that  the  current  is  con- 
centrated on  the  center  line  of  the  wire.  The  lines  of  magnetic 
induction  surrounding  a  long  straight  wire  carrying  an  electric 
current  of  which  the  return  path  is  at  a  considerable  distance, 
will  be  in  the  form  of  circles  concentric  with  the  conductor.  The 
number  of  lines,  or  flux  in  maxwells,  contained  between  any  two 
imaginary  concentric  cylinders,  of  average  radius  x  centimeters, 


80 


ELECTRIC  POWER  TRANSMISSION 


and  axial  length  I  centimeters  (see  Fig.  21)  will  be  the  product 
of  the  magnetomotive  force  by  the  permeance,  or 

4rT  .    Z  X  dx  X*t 


2irx 


10 


dx 
X  x 


where  /  is  the  current  in  the  wire,  /*  is  the  permeability,  and 
dx  is  the  separation  between  the  cylinders,  in  centimeters. 

Assuming  dx  to  become  smaller  and  smaller  without  limit, 
and  putting  p  =  1  (for  the  condition  of  flux  lines  in  ah*),  the  ex- 


Fia.  21. 

pression  for  the  total  flux  outside  the  conductor,  up  to  a  limiting 
distance  of  d  centimeters,  is 

211 


(25) 


211 ,        fd\ 

Tolog<U 


where  r  is  the  radius  of  the  conductor,  in  centimeters. 

44.  Effect  of  Taking  into  Account  the  Return  Conductor. — 
The  effective  flux  surrounding  any  single  conductor  of  a  trans- 
mission system  will  depend  upon  the  distance  of  the  parallel 
return  conductor  or  conductors. 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS   81 

Consider,  first,  the  loop  formed  by  two  parallel  conductors  of 
circular  cross-section,  one  carrying  the  outgoing  current  7  and 
the  other  carrying  the  return  current  —  /(see  Fig.  22).  The  flux 
due  to  the  current  7  in  the  conductor  A  may  be  considered  as 
extending  indefinitely  throughout  space,  with  ever-weakening 
intensity  as  the  distance  from  the  conductor  increases,  and  the 
same  argument  applies  to  the  flux  surrounding  the  return  con- 
ductor B,  the  only  difference  being  that,  if  the  direction  of  the 
flux  round  A  be  considered  positive,  that  which  surrounds  B 
will  be  in  a  negative  direction.  It  follows  that  the  whole  of  the 
magnetic  flux  due  to  the  current  7  in  A,  which  is  situated  at  a 
distance  greater  than  the  distance  d  between  centers  of  the  out- 
going and  return  conductors,  is  exactly  neutralized  by  the  flux 


FIQ.  22. — Magnetic  lines  of  force  around  two  parallel  conductors. 

due  to  the  current  -  I  in  5,  Thus,  in  Fig.  22,  it  will  be  seen  that 
the  flux  of  induction  surrounding  A  up  to  a  distance  d,  is  not 
neutralized  by  the  current  —  7  in  the  conductor  B;  but  any  mag- 
netic line,  such  as  M,  situated  at  a  greater  distance,  P,  from  the 
center  of  the  conductor  A,  is  exactly  neutralized  by  the  mag- 
netic line  N,  due  to  the  return  current  in  conductor  B,  since  it 
also  surrounds  the  conductor  A,  but  in  a  direction  opposite  to 
that  of  the  line  M.  It  follows  that  the  total  effective  flux  sur- 
rounding A — that  is,  the  resultant  flux  which  will  give  rise  to  an 
induced  e.m.f.  in  the  conductor  when  carrying  an  alternating 
current — is  merely  that  portion  of  the  total  self-produced  flux 
included  between  the  surface  of  the  conductor  and  the  surface 
of  an  imaginary  cylinder,  concentric  with  the  conductor,  and  of 
radius  d,  equal  to  the  distance  between  the  centers  of  the  out- 
going and  return  conductors.  The  formula  (25)  may,  there- 


82  ELECTRIC  POWER  TRANSMISSION 

fore,  be  used  for  calculating  the  flux  which  is  effective  in  produc- 
ing an  e.m.f  .  of  self-induction  in  a  straight  conductor  when  the 
whole  of  the  return  current  is  situated  at  a  distance  d  from  the 
center  of  the  conductor. 

45.  Effect  of  Flux  Lines  in  the  Material  of  the  Conductor.— 
The  flux  as  calculated  by  formula  (25)  is  that  which  surrounds  the 
wire,  and  when  the  current  to  which  it  owes  its  existence  alter- 
nates in  direction,  an  e.m.f.  of  self-induction  will  be  induced 
in  the  conductor. 

Since  100,000,000  lines  cut  per  second  generate  one  volt,  and 
the  total  effective  flux  surrounding  the  conductor  is  twice  created 
and  twice  destroyed  in  the  time  of  one  complete  period,  the 

w 

mean  value  of  the  e.m.f.  of  self-induction  will  be  y^  volts.     The 

virtual  or  r.m.s.  value,  on  the  sine-wave  assumption,  is-  —  •/= 

2\/2 

or  1.11  times  this  quantity,  whence 

4  44<i>f 
Induced  volts  =          ,J  (26) 


If  I  stands  for  the  virtual  value  of  the  alternating  current, 
the  maximum  value  of  <£,  by  formula  (25),  will  be 


10 

Substituting  in  formula  (26)  after  replacing  I  by  the  number  of 
centimeters  in  a  mile,  and  converting  the  Napierian  logarithms 
into  common  logarithms,  we  get 

Volts  induced  per  mile  of  conductor  =  0.00466/7  log  (-)         (27) 

This  formula  is  approximately  correct  for  conductors  of 
overhead  transmission  lines  when  these  are  of  "non-magnetic" 
material;  but  it  should  be  slightly  modified  to  take  into  account 
the  effect  of  the  flux  lines  within  the  material  of  the  conductor. 
This  additional  drop  of  pressure  is  not  easily  calculated  because 
different  amounts  of  flux  link  with  different  portions  of  the 
conductor.  It  is  obvious  that  a  portion  of  the  conductor  near  the 
surface  is  surrounded  by  fewer  flux  lines  than  a  portion  near 
the  center  of  the  cross-section.  The  result  is  that  the  e.m.f. 


ELECTRICAL  PRINCIPLES  AND  CALCULA  TIONS   83 

induced  per  unit  length  of  conductor  is  not  the  same  throughout 
the  cross-section.  This  suggests  the  possibility  of  eddy  currents 
in  the  wire;  but  what  actually  takes  place  is  a  distribution 
of  the  current  density  over  the  cross-section  such  that  the  total 
impedance  drop  —  or  the  IR  drop  added  (vectorially)  to  the 
IX  drop  —  will  have  the  same  value  at  all  parts  of  the  conductor 
cross-section.  The  correct  calculation  of  the  internal  reactance 
drop  for  "non-magnetic"  cylindrical  conductors  is  given  in 
Prof.  H.  B.  Dwight's  book  on  Transmission  Line  Formulas. 

The  result  is  that  the  inductive  pressure  drop  is  actually 
somewhat  greater  than  as  indicated  by  formula  (27),  which 
neglects  the  internal  flux.  The  corrected  formula  is 


0.000506 


]    (28) 


The  antilogarithm  of  the  constant  in  the  brackets  is  1.285,  and 

it  is  more  convenient  to  write  the  formula 

Reactive  voltage  drop  per  mUel   _  ,         /A 

of  single  conductor  J  \  rl 

which  is  the  same  as  formula  (6)  already  given  in  Chapter  II. 
The  reactance  of  stranded  cables  is  slightly  less  than  that  of 
solid  conductors  of  the  same  cross-section,  owing  to  the  fact  that 
the  overall  diameter  of  the  cable  is  greater  than  that  of  the  solid 
wire. 

Excellent  tables  giving  inductive  reactance  in  ohms  per  mile 
for  different  spacings  and  sizes  of  wires  are  given  in  the  Stand- 
ard Handbook  for  Electrical  Engineers;  these  figures,  when  mul- 
tiplied by  the  value  of  the  current  flowing  in  the  conductor, 
give  the  induced  volts  as  calculated  by  formula  (29). 

46.  Iron  as  a  Material  for  Transmission  Line  Conductors.  — 
The  European  war,  by  limiting  the  supply  of  copper  and  aluminum 
available  in  Germany,  and  by  causing  an  abnormal  increase  in  the 
price  of  these  metals  all  over  the  world,  has  led  electrical  engi- 
neers to  consider  the  possibility  of  using  other  metals  as  conductors 
of  electricity.  Zinc  has  been  used  in  Germany  for  insulated 
wires  and  cables;  but  it  is  mechanically  weak,  and  generally 
unsuitable  for  overhead  transmission  lines. 

When  considering  the  economic  advantages  of  using  iron  or 
steel  conductors,  it  is  necessary  to  take  into  account:  (a)  the 
cost  of  the  material  at  the  place  where  it  is  to  be  used  ;  (6)  the 


84  ELECTRIC  POWER  TRANSMISSION 

"life"  of  galvanized  iron  wires  or  cables  as  compared  with 
that  of  copper  and  aluminum;  (c)  the  energy  losses  in  trans- 
mission; (d)  the  voltage  regulation,  and  the  increased  cost  (if 
any)  of  maintaining  the  pressure  within  specified  limits  at  the 
receiving  end  of  the  line. 

Under  item  (c)  the  greatly  increased  "skin  effect"  with  alter- 
nating currents  must  be  taken  into  account  as  well  as  the  higher 
specific  resistance  which  requires  a  larger  cross-section  of  iron 
than  of  copper  wire  even  when  the  transmission  is  by  continuous 
currents. 

Under  item  (d)  the  internal  inductance  of  the  wire — which  is 
almost  negligible  with  copper  or  aluminum — becomes  a  matter  of 
considerable  importance  owing  to  the  greatly  increased  magnetic 
flux  in  the  material  of  the  conductor  when  iron  or  steel  is  used. 

Although  cables  of  extra  high  strength  steel  wire  are  occasion- 
ally used  for  long  spans — such  as  river  crossings — on  important 
overhead  lines  transmitting  large  amounts  of  energy,  this  material 
would  not  be  satisfactory  as  a  substitute  for  copper  or  aluminum 
except  on  comparatively  short  sections  of  the  entire  line.  It 
seems,  however,  that  iron  or  steel  conductors  may  be  used  to 
advantage  on  short-distance  small-power  transmissions  when  the 
price  per  pound  of  copper  wire  has  been  forced  up  to  30  cents  or 
more. 

On  account  of  the  wide  variations  in  the  electric  and  magnetic 
qualities  of  the  different  grades  of  iron  and  steel  wire,  it  is  prac- 
tically impossible  to  predetermine  losses  and  pressure  regulation 
with  a  high  degree  of  accuracy.  The  particulars  and  data, 
together  with  the  numerical  example,  in  the  following  articles 
should,  however,  be  helpful  to  the  reader  when  making  prelimi- 
nary calculations  on  iron  wire  transmission  lines. 

47.  Apparent  Resistance  of  Iron  and  Steel  Conductors. — 
The  relative  resistances  of  iron  and  copper  wires  were  given  in 
Article  41,  and  wire  tables  will  be  found  in  the  Handbooks  for 
Electrical  Engineers;1  but  the  accompanying  table  includes 
the  sizes  likely  to  be  used  in  practice.  The  figures  give  the 
approximate  resistance  to  continuous  currents  and  must  be  multi- 
plied by  the  skin  effect  factor  when  the  current  is  alternating. 

1  Very  complete  particulars  relating  to  conductor  materials  will  be  found 
in  the  Handbook  on  Overhead  Line  Construction  published  by  the  National 
Electric  Light  Association. 


ELECTRICAL  PRINCIPLES  AND  CALC ULA  TIONS   85 


APPROXIMATE  RESISTANCE  PER  MILE  OF  SOLID  GALVANIZED  IRON  WIRE  AT 
68°  F. 


Gauge  No., 
B.  W.  G. 

Diameter,  in. 

Ohms  per  mile 

Weight 
lb.  per  mile 

E.  B.  B. 

B.  B. 

2 

0.284 

4.1 

4.9 

1160 

3 

0.259 

4.9 

5.8 

960 

4 

0.238 

5.8 

6.9 

810 

5 

0.220 

6.8 

8.1 

690 

6 

0.203 

8.0 

9.5 

590 

7 

0.180 

10.2 

12.1 

460 

8 

0.165 

12.1 

14.4 

390 

9 

0.148 

15.0 

17.9 

315 

10 

0.134 

18.2 

21.7 

260 

7-strand  Ke-in.  galv.  steel  (ordinary) .  5.4  ohms  per  mile.  .  1110  lb.  per  mile 
7-strand  y±-\n.  galv.  steel  (ordinary) .  .  8.6  ohms  per  mile.  .  660  lb.  per  mile 
7-strand  Ke-in-  Siemens-Martin  steel.  7.4  ohms  per  mile.  .  1110  lb.  per  mile 
7-strand  ^-in.  Siemens-Martin  steel. .  9.6  ohms  per  mile. .  660  lb.  per  mile 
7-strand  K-in.  E.  E.  B.  iron 7.8  ohms  per  mile .  .  660  lb.  per  mile 

The  resistance  of  ordinary  steel  wire  is  about  30  per  cent,  higher 
than  that  of  the  E.  B.  B.  iron. 

The  skin  effect  coefficient  will  depend  not  only  upon  the  diame- 
ter of  the  wire  and  the  frequency,  but  also  upon  the  resistivity 
and  magnetic  properties  of  the  iron  or  steel.  The  magnetic 
permeability  will,  in  its  turn,  be  some  function  of  the  current 
in  the  wire,  and  it  is  not  possible  to  express  the  skin  effect  co- 
efficient (k)  by  means  of  a  simple  formula  as  was  done  in  Article 
42  in  connection  with  "non-magnetic"  conductors.  The  co- 
efficient k,  for  a  frequency  /  =  60,  as  calculated  from  tests  on 
certain  samples  of  iron  and  steel  conductors,  may  be  obtained 
from  Fig.  23. 

When  the  true  ohmic  resistance,  R,  of  the  iron  conductor,  is 
multiplied  by  the  skin  effect  factor  (A;),  the  product,  R',  will 
be  the  effective  resistance  of  the  wire  to  an  alternating  current 
of  the  given  frequency  (in  this  case  60  cycles  per  second).  In 
other  words,  if  the  power  wasted  in  heating  the  wire  with  con- 
tinuous currents  is  PR,  it  will  be  72  (kR)  when  carrying  an 
alternating  current  of  virtual  value  7. 

48.  Internal  Reactance  of  Iron  and  Steel  Conductors. — The 
formula  (28)  in  Article  45  gives  the  total  inductive  voltage  drop 


86  ELECTRIC  POWER  TRANSMISSION 

in  a  mile  of  "non-magnetic"  conductor;  the  term  0.000506/Z 
being  the  pressure  drop  due  to  the  flux  lines  within  the  material 
of  the  conductor.  Obviously,  if  the  permeability  is  no  longer 
H  =  1,  but  a  larger  number,  the  loss  of  pressure  will  be  greater, 
and  this  is  what  occurs  with  iron  conductors. 

It  is  a  very  simple  matter  to  write 

Voltage  drop  per  mile  of  single  conductor  1    _  ft  nnnsnA/7  v 
due  to  internal  reactance  /  = 

2.6 


01        23        4567        8        9       10      11      12      13      14 

Currents  Amperes, 

FIG.  23. — Curves  giving  skin  effect  coefficient  for  iron  conductors  for  a  frequency 
/  =60. 

but  in  this  connection  n  is  a  purely  imaginary  number  repre- 
senting an  "equivalent  permeability"  which  cannot  be  calcu- 
lated, and  which  is,  in  any  case,  some  function  of  the  current  (/) 
and  the  frequency  (/).  A  formula  in  this  shape  is  therefore 
practically  worthless,  and  it  is  necessary  to  rely  on  test  data 
obtained  from  sizes  and  grades  of  wire  approximating  to  those 
of  the  conductor  it  is  proposed  to  use. 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS   87 


One  of  the  most  valuable  contributions  available  for  the  use 
of  those  desiring  to  calculate  the  probable  regulation  and  losses 
in  lines  using  iron  or  bi-metallic  conductors,  is  the  Paper  No.  252 
by  J.  M.  Miller  issued  by  the  Bureau  of  Standards  at  Washington, 
D.  C.  Additional  data  will  be  found  in  the  article  by  Messrs. 
C.  E.  Oakes  and  W.  Eckley  published  in  the  Electrical  World  of 
Oct.  14,  1916,  in  the  article  by  L.  W.  W.  Morrow  in  the  Electrical 
World  of  July  14,  1917,  and  in  the  article  by  C.  E.  Oakes  and  P. 


.-270 

a 

I 

i* 


Approximate  Vajnei  of 
Internal  Beactance  Voltage 
Drop  (  Vi  )  In  Iron  and  Steel 
Conductors   carrying  Alter- 
nating Current!  at  /.=  .60. 


2       a       i       5       6        7       8        9       10      11      12      13      14      15      16      17 


FIG.  24. — Internal  reactive  voltage  drop  in  iron  conductors. 

A.  B.  Sahm  in  the  Electrical  World  of  July  27,  1918.  Data 
from  these  sources  have  been  used  in  preparing  the  curves  of 
Figs.  23  and  24.1 

For  calculating  the  total  inductive  voltage  drop  in  an  iron  wire 
transmission  line,  a  modification  of  formula  (28)  may  conven- 
iently be  used,  because  it  is  desirable  to  distinguish  between  the 
external  reactance — which  depends  only  upon  the  size  and  spacing 
of  the  conductors,  apart  from  the  material — and  the  internal 

1  A  valuable  collection  of  data  referring  to  iron  wires  for  transmission  lines 
will  be  found  in  Prof.  W.  T.  Ryan's  article  "Iron  wire  for  short  high-voltage 
lines"  in  the  Electrical  Review  (Chicago)  Sept.  22,  1917,  Vol.  71,  p.  496. 


88  ELECTRIC  POWER  TRANSMISSION 

reactance  which  will  be  greater  with  "magnetic  "  than  with  "non- 
magnetic" materials. 

If  L  is  the  external  inductance  (coefficient  of  self-induction), 
in  henrys,  of  the  conductor  due  to  the  flux  of  induction  outside 
the  wire,  and  L»  is  the  internal  inductance  due  to  the  flux  of 
induction  inside  the  wire,  the  total  reactance  (in  ohms)  being 

X  (total)  =  X (external)  +  X (internal) 
may,  on  the  sine  wave  assumption,  be  written 

X  (total)  =  27T/L  +  27r/Li 
and  the  reactive  drop  (in  volts)  when  the  current  is  /  amperes  is 

IX    =    27T//L   +   27T//L, 

The  value  of  L  per  mile  of  conductor  is  0.000741  log  -»  whence 

Total  reactive  drop  in  volts  ]  , 

per  mile  of  single  iron  or  \  =  0.00466//  log  -  -f  Vt         (30) 
steel  conductor  J 

where  F<  =  2-jrfFLi  and  has  to  be  determined  experimentally. 
Its  value,  for  a  frequency  /  =  60,  may  be  read  off  Fig.  24  which, 
however,  refers  only  to  a  limited  number  of  sizes  and  kinds  of 
wire. 

49.  Example  of  Calculations  for  Iron  Wire  Conductors. — Given 
a  transmission  line  10  miles  long  consisting  of  No.  4  E.  B.  B.  galva- 
nized iron  wires  spaced  3  feet  apart,  carrying  a  current  of  5 
amperes  at  a  frequency  of  60;  calculate  (a)  the  loss  of  power,  and 
(6)  the  loss  of  pressure,  in  each  wire. 

(a)  The  D.C.  resistance  (from  wire  table)  is  R  =  5.8  X  10 
=  58  ohms. 

The  skin  effect  factor  (from  Fig.  23)  is  k  =  2.2 
whence  R'  =  58  X  2.2  =  127.5  ohms 

The  watts  lost  =  PR'  =5  X  5  X  127.5  =  3.19  k.w.  per  wire. 

(6)  The  internal  reactive  pressure  drop  per  mile  (from  Fig.  24) 
is  Vi  =  37  volts,  whence  IX  (total)  for  10  miles  of  wire  (by 
formula  (30)  is 

10  X  0.00466  X  60  X  5  X  log  ^r^  +  370  =  (say)  405  volts 

u.  i  iy 

The  IRf  drop  being  5  X  27.5  =  638  volts,  it  follows  that  the 
impedance  drop  is 

IZ  =  V(638)s+  (405) 2 
=  755  volts 


ELECTRICAL  PRINCIPLES  AND  CALC ULA  TIONS   89 

This  figure  does  not,  however,  necessarily  represent  the  differ- 
ence in  pressure  between  the  generating  and  receiving  ends  of  the 
line;  but  this  point  will  be  taken  up  in  the  following  article. 

50.  Inherent  Regulation  of  Transmission  Line.  Regulation 
Diagrams. — The  fundamental  vector  diagram,  Fig.  10,  which 
was  described  in  Article  9  of  Chapter  II,  is  reproduced  here  for 
convenience.  The  resistance  drop  CB  (of  which  the  numerical 
value  was  638  volts  in  the  foregoing  example)  is  drawn  parallel 
to  the  current  vector,  while  the  reactance  drop  DC  (of  which  the 
numerical  value  was  405  volts  in  the  example)  is  drawn  at  right 
angles  to  the  current  vector.  The  impedance  drop  is  DB, 


0  (I)  A 

Fia.  10. — Vector  diagram  for  line  calculations — capacity  neglected. 

but  this  does  not  correspond  to  the  loss  of  pressure  in  trans- 
mission except  when  the  angle  <f>  happens  to  have  the  same  value 
as  the  angle  9.  The  difference  in  pressure  between  the  genera- 
ting end  and  receiving  end  voltages  maybe  calculated  as  explained 
in  Article  9,  and  it  will  depend  not  only  on  the  resistance  and 
size  and  spacing  of  the  conductors,  but  also  on  the  power  factor 
of  the  load,  since  this  will  determine  the  position  of  the  point 
B  on  the  dotted  circle.  It  is  the  position  of  the  point  B  on  the 
circle  that  modifies  the  ratio  of  the  length  FD  to  the  length  BD, 
even  if  the  proportions  of  the  impedence  triangle  BCD  remain 
unaltered.  Problems  can  be  solved  graphically  by  drawing 
the  diagram,  Fig.  10,  to  the  proper  scale;  but  the  objection  to 
this  method  is  that  the  radius  OB  is  generally  large  in  proportion 


90 


ELECTRIC  POWER  TRANSMISSION 


to  the  quantities  represented  by  the  impedance  triangle,  and  the 
process  is  either  tedious  or  the  results  are  unsatisfactory.  The 
field  for  ingenuity  in  the  construction  of  practical  charts  based 


.1 

Power  Factor 


1.0        6*      10,?      15?     20?     25?     30*    35% 
Curved  Lines-  -Percentage  Voltage  Loss 


FIG.  25. — Mershon  diagram  for  determining  voltage  regulation. 

on  the  fundamental  diagram  (Fig.  10)  is  very  great.  One  of  the 
methods  of  obtaining  graphical  solutions  is  with  the  aid  of  the 
Mershon  diagram. 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS   91 

In  Fig.  25  curves  concentric  with  the  dotted  circles  of  Fig.  10 
are  drawn  on  a  piece  of  squared  paper  from  a  center  which  lies 
on  the  prolongation  of  the  base  line,  but  at  a  considerable  dis- 
tance outside  the  diagram.  The  radius  of  the  inner  circle  is  10 
(or  100)  divisions  in  length,  and  the  projection  on  the  horizontal 
axis  of  any  point  B  is  therefore  the  cosine  of  the  angle  BOA  of 
Fig.  10  and  it  indicates  directly  the  power  factor  at  the  receiving 
end.  By  expressing  the  calculated  resistance  and  reactive 
voltage  drops  as  percentages  of  the  receiving  end  pressure,  the 
impedance  triangle  can  readily  be  drawn  to  the  proper  scale, 
and  by  making  the  space  between  the  circles  equal  to  the  side 
of  the  squares  on  the  divided  paper,  the  regulation,  or  difference 


FIG.   26. — Vector   diagram   illustrating   approximate   method   of   determining 
regulation. 

between  generating  and  receiving  end  pressures  (FD  hi  Fig.  10), 
can  be  read  off  the  diagram  as  a  percentage  of  the  receiving  end 
pressure. 

As  an  example  of  the  use  of  the  diagram,  suppose  the  power 
factor  of  the  load  is  0.77,  and  that  the  calculated  components  of 
the  pressure  drop  are, 

Resistance  volts  =  17  per  cent,  of  receiving  end  pressure. 

Reactance  volts  =  22  per  cent,  of  receiving  end  pressure. 

From  the  division  on  the  horizontal  axis  corresponding  to 
power  factor  0.77  follow  the  vertical  ordinate  until  it  meets  the 
inner  circle  at  B;  then  measure  horizontally  17  divisions,  and 
vertically  22  divisions,  and  the  point  D  which  lies  on  the  dotted 


92  ELECTRIC  POWER  TRANSMISSION 

circle  27.5  divisions  larger  in  radius  than  the  inner  circle  (which  is 
described  with  a  radius  equal  to  100  divisions)  indicates  that  the 
difference  in  pressure  between  generating  and  receiving  ends 
of  the  line  is  27.5  per  cent,  of  the  receiving  end  pressure. 

Consider  now  Fig.  26,  which  is  merely  a  repetition  of  the 
fundamental  diagram,  Fig.  10,  with  the  addition  of  a  few  lines. 
Drop  the  perpendicular  DM  on  the  radius  OB  extended  beyond 
the  point  B.  It  will  be  seen  that  when  the  angle  DOB  is  small, 
that  is  to  say,  when  there  is  little  difference  between  the  power 
factors  at  the  receiving  and  generating  ends  of  the  line,  the  dis- 
tance M N  will  be  very  small,  and  for  nearly  all  practical  pur- 
poses the  voltage  regulation  may  be  expressed  by  the  ratio 

•0£-  instead  of  ^g>  this  last  being  theoretically  correct  and  as 

given  by  the  Mershon  diagram.  By  adopting  the  alternative 
construction,  and  replacing  the  arc  DN  by  a  straight  line  per- 
pendicular to  either  OD  or  OB,  the  necessity  for  drawing  circles 
from  a  center  outside  the  limits  of  a  practical  diagram  is  avoided. 

The  method  used  by  Professor  L.  A.  Herdt  for  the  calculation  of 
transmission  lines  (originally  described  in  the  Electrical  World  of 
Jan.  2,  1909)  employs  this  approximation;  and  it  is  also  employed 
in  the  method  about  to  be  described,  which  the  writer  has  found 
very  quick  and  convenient  for  practical  calculations. 

It  will  be  observed  that  if  the  impedance  triangle  BCD  (Fig. 
26)  be  moved  round  on  the  point  B  through  an  angle  0,  so  that 
the  hypotenuse  BD  now  occupies  the  position  BDi,  the  perpen- 
dicular dropped  from  DI  on  the  extension  to  the  horizontal  line 
BC,  meets  this  line  at  the  point  MI,  the  distance  BM\  being 
obviously  equal  to  BM.  Thus,  by  revolving  the  impedance 
triangle  through  an  angle  6  such  that  cos  6  =  the  power  factor 
of  the  load,  the  projection  of  the  hypotenuse  on  any  line  parallel 
to  the  current  vector  will  be  a  measure  of  the  volts  lost  in 
transmission. 

To  apply  this  method  in  practice,  nothing  more  is  required 
than  a  piece  of  squared  paper  and  a  piece  of  tracing  paper.  The 
squared  paper  is  divided  into  any  convenient  number  of  equal 
parts  to  represent,  horizontally,  the  percentage  ohmic  drop 
of  voltage,  and,  vertically,  the  percentage  reactive  voltage  drop, 
as  indicated  in  Fig.  27.  On  the  vertical  axis  on  the  left-hand  side 
of  the  diagram,  a  power  factor  scale  is  provided.  This  is  merely 
an  arbitrary  length  divided  into  ten  equal  parts  with  suitable  sub- 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS   93 

divisions  so  chosen  as  to  make  use  of  the  horizontal  ruling  of  the 
squared  paper.  This  scale  is  used  for  turning  the  hypotenuse  of 
the  impedence  triangle  through  the  proper  angle,  as  will  be  ex- 
plained shortly. 

The  method  of  using  the  diagram  is  best  explained  by  working 


p 

( 

- 

25 
23 
21 
19 

"I 
»J 

13  a 
llj 
S  g 

5 
3 
1 

D 
<> 

- 

> 

- 

0.9 
0.8 
0.7 
0.6 
0.5 

<* 

- 

- 

- 

•     —  - 





__  — 











_^-«— 

- 

- 

-.    - 

(¥» 

- 

-•- 

D, 

- 

- 

Pe 

rcent 

ige  C 

hnric 

Volta 

ere  Dr 

>p.   i 

and: 

iesul 

ant) 

- 

n 

3 

6 

7 

9 

11 

13 

15 

17 

19 

21 

23 

25 

27 

29 

"""    "™< 



•—  .   —. 

-.. 



—  ••<•. 

-—     — 

. 

• 







FIG.  27. — Author's  diagram  for  determining  voltage  regulation. 

out  an  example.     The  data  used  for  illustrating  the  Mershon 

chart  will  be  suitable: 

Power  factor  of  load  =  cos  6  =  0.77 
Ohmic  volts  =17  per  cent. 

Reactive  volts  =  22  per  cent. 


94  ELECTRIC  POWER  TRANSMISSION 

Place  a  piece  of  tracing  paper  over  the  diagram  (Fig.  27)  and 
mark  upon  it  the  point  D,  17  divisions  to  the  right  of  the  vertical 
axis,  and  22  divisions  above  the  horizontal  axis.  Then,  with  a 
pin  or  pencil  point  held  at  the  point  0,  move  the  tracing  paper 
through  an  angle  of  39  degrees  40  minutes  (cos  39°  40'  =  0.77), 
bringing  the  point  D  to  D\,  and  read  on  the  horizontal  axis  the 
distance  27.1,  which  is  the  difference  between  the  pressures  at 
the  two  ends  of  the  line,  expressed  as  a  percentage  of  the  receiving- 
end  pressure.  The  result,  as  read  off  the  Mershon  diagram  was 
27.5,  which  might  at  first  sight  be  thought  to  be  more  nearly 
correct;  but,  as  a  matter  of  fact,  the  writer's  method  will  usually 
give  more  accurate  results  notwithstanding  that  the  solution  is 
not  theoretically  correct.  This  is  because  the  impedance  triangle 
is  very  much  larger  for  the  same  size  of  chart  than  in  the  Mer- 
shon diagram;  and  the  subdivisions  are  more  easily  read. 
When  the  point  Di  falls  between  the  two  inclined  dotted  lines 
drawn  on  the  diagram  (Fig.  27),  this  is  an  indication  that  the 
error  introduced  by  substituting  the  chord  for  the  arc  is  less  than 
half  of  1  per  cent. 

The  use  of  the  power  factor  scale  will  now  be  explained.  It 
is  not  necessary,  as  suggested  in  working  out  the  example,  to 
calculate  the  angle  0  from  the  value  of  the  power  factor  and  then 
set  out  this  angle  on  the  diagram.  If  in  addition  to  marking  the 
point  D  on  the  tracing  paper,  the  position  of  the  point  P  is  also 
marked,  it  is  merely  necessary  to  move  the  tracing  paper  round 
(on  the  center  0)  until  the  point  P  falls  on  the  horizontal  line  rep- 
resenting the  required  power  factor,  as  this  will  ensure  that  the 
point  D  has  been  moved  through  the  proper  angle.  The  reason 
of  this  will  be  obvious  to  anyone  possessing  even  the  most  ele- 
mentary knowledge  of  trigonometry. 

If  it  is  preferred  to  work  with  trigonometrical  tables,  the 
formulas  (7)  to  (11)  of  Article  9,  Chapter  II  may  be  used  instead 
of  the  diagram  Fig.  25  or  27. 

51.  Pressure  Available  at  Intermediate  Points  on  a  Transmis- 
sion Line. — Referring  again  to  Fig.  10,  the  volts  per  phase  at  gen- 
erating end  are  Vn  and  at  receiving  end,  En,  the  total  drop  being 
(Vn  —  En}  volts.  It  does  not  follow,  however,  that  the  pressure 

available  at  a  point  half  way  along  the  line  will  be  Vn  —  (— ^ — ~) 

because  the  power  factor  is  rarely  the  same  at  all  points. 

The  method  of  calculating  the  pressure  available  at  some  inter- 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS   95 

mediate  point  I/  miles  from  the  supply  station  on  a  line  of 
total  length  L  miles,  when  the  effects  of  capacity  are  negligible, 

,T  /. 

is  illustrated  in  Fig.  28  where  C'C  =  BC  (-~\  and  OD  —  OD' 

is  a  measure  of  the  voltage  drop  between  the  supply  end  and  the 
point  considered.  The  power  factor  angle  at  this  point  will 
be  ^  which  can  be  calculated  by  making  the  required  changes 
in  the  formulas  of  Article  9.  Thus,  formula  (10)  would  be  written 
BC'  +  En  cos  0 


OD'  = 


cos  \l/ 


FIG.  28. — Vector  diagram  showing  pressure  at  intermediate  point  on  trans- 
mission line. 

and  the  procedure  throughout  is  exactly  the  same  as  if  calculat- 
ing the  required  generating  end  voltage  on  a  line  (L  —  L') 
miles  in  length,  to  give  En  volts  at  the  receiving  end  when  de- 
livering 7  amperes  at  a  power  factor  cos  0. 

Such  calculations  are  usually  unnecessary  refinements,  and 
the  error  introduced  by  assuming  a  uniform  drop  of  pressure 
along  the  line  is  rarely  of  any  practical  importance. 

The  manner  in  which  the  values  of  pressure  drop,  as  calculated 
for  a  single  conductor,  are  used  in  determining  the  inherent 
regulation  of  three-phase  lines  was  explained  in  Article  11  of 
Chapter  II.  The  principles  governing  the  inductive  effects 
with  any  number  and  arrangement  of  parallel  conductors,  are 
discussed  in  Appendix  I. 


96  ELECTRIC  POWER  TRANSMISSION 

52.  Capacity  of  Transmission  Lines.  —  The  formula  given  in 
Chapter  II  (Article  10)  for  the  capacity  in  microfarads  per  mile 
of  conductor,  as  measured  between  wire  and  neutral  was: 


C.  -  (12) 


This  formula  is  not  theoretically  correct  and  would  not  be 
applicable  if  the  distance  d  were  very  small  in  relation  to  the 
diameter  of  the  wire  (or  the  radius  r);  but  for  overhead  trans- 
missions it  is  a  serviceable  formula,  and,  in  the  writer's  opinion, 
it  may  be  used  in  all  practical  calculations.  The  question  'of 
capacity  on  overhead  lines  is,  however,  one  of  very  great  impor- 
tance, especially  in  view  of  the  increasing  pressures  and  distances 
of  transmission  ;  and  it  is  felt  that  some  space  should  be  devoted 
to  it,  even  if  it  be  only  to  sum  up  our  present  knowledge  on  this 
subject,  and  refer  the  reader  to  sources  from  which  he  can 
obtain  more  complete  information. 

The  exact  formula,1  which  gives  the  linear  capacity  in  micro- 
farads per  mile  between  two  cylindrical  parallel  wires  is 

c  °-0194  ,3n 

log  (a  +  V^^l] 
where  a  =   ~-  ;  but  it  is  more  generally  useful  to  consider  the 

capacity  as  being  measured  between  one  wire  and  the  neutral 
potential  surface.  This  will  be  twice  the  value  of  the  capacity  as 
measured  between  the  two  wires  ;  but,  when  calculating  the  charg- 
ing current,  it  is  the  voltage  between  wire  and  neutral  surface  that 
must  be  taken,  if  this  latter  value  of  the  capacity  is  used. 

The  formula  (31)  may  be  put  into  another  form  which  is  very 
convenient  if  tables  of  hyperbolic  functions  are  available. 

In  the  formula  (31)  common  logarithms  are  referred  to  in  the 
denominator;  but  by  making  the  proper  correction  to  the  numer- 
ator and  substituting  Napierian  logs,  the  denominator  becomes 
log,  (a  +  \/«2  —  1)  which  is  the  quantity  of  which  the  hyperbolic 
cosine  is  a.  Thus,  the  inverse  hyperbolic  cosine  of  a,  or  cosh-1a, 
is  the  equivalent  of  loge  (a  +  V«2  —  1);  and  with  the  corrected 
numerator,  the  formula  (31)  becomes, 

C     -    °-°447  (32) 

Lm~  cosh-1  a 

1  H.  Fender  and  H.  S.  Osborne  in  Electrical  World,  Sept.  22,  1910,  p.  667. 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS    97 

If  the  capacity  per  mile  of  single  conductor,  measured  between 
wire  and  neutral,  is  required,  the  numerators  of  these  formulas 
must  be  doubled,  and  the  correct  formula  may  be  written  either 

0.0388 


log  (a  +  Va2  -  1) 
or, 

0.0895 

C/  jn      *  ^ 

cosh"1 a 


(34) 


Some  excellent  practical  diagrams  based  on  these  formulas 
are  to  be  found  in  an  article  by  Dr.  A.  E.  Kennelly  which  ap- 
peared in  the  Electrical  World  of  Oct.  27,  1910. 

The  approximate  formula  (12)  given  in  Chapter  II  may  be 
written 

=  0.0388 
m  ~  log  2a 

and,  by  comparing  this  with  the  correct  formula  (33),  it  will  be 
seen  that  the  first  gives  results  slightly  smaller  than  the  true 
values;  but  when  a  is  large,  that  is  to  say,  when  the  distance  be- 
tween wires  is  many  times  the  diameter,  the  error  is  negligible. 
The  error  only  becomes  appreciable  if  a  is  less  than  10,  and  even 
if  a  is  as  small  as  4  (a  quite  impossible  state  of  things  on  an 
overhead  transmission  with  bare  wires),  the  error  would  be  only 
0.8  per  cent. 

53.  Capacity  of  Three-phase  Lines. — The  formulas  in  the  last 
article  give  the  capacity  between  two  parallel  wires  as  measured 
from  wire  to  neutral,  and  in  the  case  of  a  single-phase  transmis- 
sion, the  capacity  between  the  two  wires  would,  as  it  were,  consist 
of  two  such  capacities  in  series,  and  would  therefore  measure 
half  the  value  given  by  these  formulas,  all  as  previously  men- 
tioned. It  should,  however,  be  noted  that  it  makes  no  difference 
which  value  of  the  capacity  is  taken  for  the  purpose  of  calculating 
the  charging  current,  provided  proper  attention  is  paid  to  the 
potential  difference  available  for  charging  the  condenser.  In 
the  case  of  the  single-phase  transmission,  the  pressure  available 
for  charging  the  two  imaginary  condensers  in  series,  is  exactly 
twice  the  pressure  between  one  wire  and  neutral. 

Consider,  now,  a  three-phase  transmission  with  the  conductors 
occupying  the  vertices  of  an  equilateral  triangle,  as  indicated  in 


98  ELECTRIC  POWER  TRANSMISSION 

Fig.  29.  If  the  radius  r  of  the  conductors  and  the  distance  d 
between  them  are  the  same  as  in  the  case  of  a  single-phase 
transmission,  then  the  capacity  as  measured  between  the  wire  and 
neutral  is  the  same  for  the  three-phase  as  for  the  single-phase 
transmission;  but  the  charging  current  is  different  because  the 
potential  difference  across  each  imaginary  condenser  is  no  longer 

Tjl  Tjl 

—  but  — /=>  where  E  stands  for  the  voltage  between  wires.  By 
2  v3 

treating  the  three-phase  system — or  indeed  any  polyphase 
system — as  a  combination  of  several  single-phase  systems  each 
having  a  condenser  connected  between  conductor  and  ground, 


FIG.  29. — Distribution  of  capacity.     Three-phase  transmission. 

the  calculation  of  capacity  currents  becomes  a  comparatively 
simple  matter,  unless  great  refinements  and  scientific  accuracy 
are  aimed  at. 

It  has  been  shown  by  Mr.  Frank  F.  Fowle1  and  other  careful 
investigators  in  this  field,  that  the  presence  of  the  conducting 
ground  or  other  neighboring  circuits  affects  only  very  slightly 
the  capacity  between  the  conductors  of  an  overhead  transmission. 
By  systematic  transposition  of  wires  on  a  long  transmission,  so 
that  each  conductor  occupies  the  same  position  relatively  to 
ground  and  neighboring  parallel  wires  over  the  same  portion  of 
the  total  distance,  even  these  slight  unbalancings  of  the  charging 
currents  can  be  corrected  if  desired. 

The  electrostatic  capacity  of  underground  cables — in  which  the 
conductors  are  not  only  very  close  together,  but  are  separated  by 

1 "  The  Calculation  of  Capacity  Coefficients  for  Parallel  Suspended  Wires," 
Electrical  World,  Aug.  12,  19,  and  26,  1911. 


ELECTRICAL  PRINCIPLES  AND  CALCULA  TIONS   99 

insulating  materials  of  which  the  dielectric  constant  is  no  longer 
unity  as  in  the  case  of  air — will  be  considered  in  Chapter  VII. 

54.  Charging  Current  Due  to  Capacity  of  Transmission  Lines. 
— Although  on  short,  low-voltage,  lines  the  charging  current  (or 
capacity  current)  is  so  small  as  to  be  negligible,  this  current 
becomes  a  matter  of  considerable  importance  on  long-distance 
high-voltage  transmission  lines.  In  Article  10  of  Chapter  II 
the  charging  current  as  calculated  for  an  existing  overhead 
transmission  amounted  to  42.4  amperes,  representing  an  output 
of  7350  apparent  kilowatts  from  the  generating  station  with  the 
line  entirely  disconnected  from  the  load  at  the  receiving  end. 

Assuming  a  sine  wave  of  impressed  e.m.f.,  it  is  easy  to  calcu- 
late the  charging  current  of  a  condenser  of  known  capacity.  The 
fundamental  law  of  the  dielectric  circuit  is 

*   =  #(max.)    X  C  (35) 

where  ^  is  the  maximum  value  of  the  dielectric  flux  expressed 
in  coulombs;  #(max.)  is  the  maximum  value  of  the  alternating 
voltage;  and  C  is  the  capacity  (or  permittance)  of  the  condenser, 
expressed  in  farads. 

The  charge,  or  quantity,  of  electricity — i.e.,  the  dielectric  flux 
— will  reach  its  maximum  value  (^)  at  the  instant  when  the 
charging  current  is  changing  its  direction,  that  is  to  say,  when 
the  current  is  zero,  and  since  quantity  of  electricity  =  current 
X  time,  we  may  write  SF  =  average  value  of  charging  current  (in 
amperes)  during  one  quarter  period  X  time  (in  seconds)  of  one 
quarter  period 

*  IcX  ^ 


where  Ic  stands  for  the  virtual  or  r.m.s.  value  of  the  charging 
current,  on  the  sine  wave  assumption.  Let  E  stand  for  the  vir- 
tual value  of  the  voltage  across  the  condenser  of  capacity  C 
farads,  then  $(max.)  —  \/2  E,  and  formula  (35)  becomes 


whence 

Ic  =  ZwfEC  (36) 

which  is  the  well-known  formula  for  calculating  capacity  current 
when  sinusoidal  wave  shapes  are  assumed.  This  is  the  same  as 
formula  (13)  of  Chapter  II,  which  was  given  without  proof. 


100  ELECTRIC  POWER  TRANSMISSION 

It  is  possible  to  express  the  charging  current  on  overhead  lines 
in  terms  of  the  external  inductance,  or  of  the  external  reactive 
voltage  drop.  This  is  due  to  the  fact  that  there  is  a  constant 
relation  between  the  inductance  and  the  capacity,  which  is 
independent  of  the  size  and  spacing  of  the  conductors.  Thus, 
the  formula  for  capacity  in  microfarads  per  mile  (page  29)  is 

(12) 


while  the  formula  for  the  external  inductance  per  mile  (page  88) 
is 

L  =  0.000741  log  ^ 
giving  a  constant  product 

CmL  =  34700  (37) 

The  external  reactive  pressure  drop  is 
(IX)  =  27T/L 

whence  a  value  for  L  in  terms  of  reactive  drop  is  obtained.     This 
value,  substituted  in  (37)  gives 

Cm  =  34,700  (IX) 
By  putting  this  value  of  Cm  in  formula  (36),  we  get, 


34,700  (IX)  X  106 

W17 

(39) 


8.78  (IX)  X  10- 

These  formulas  for  calculating  the  magnitude  of  the  charging 
current,  when  multiplied  by  the  length  of  the  line  in  miles, 
will  give  the  charging  current  entering  the  line  at  the  generat- 
ing end.  The  result  is  usually  smaller  than  the  value  obtained 
by  measurement  on  actual  lines.  The  reason  is  that  the  assump- 
tion of  sinusoidal  impressed  e.m.f.  is  rarely  justified,  and  the 
irregularities  and  peaks  in  the  actual  pressure  wave  may  cause 
an  increase  of  charging  current  amounting  to  20  or  even  40  per 
cent,  of  the  calculated  value.  These  considerations  emphasize 
the  absurdity  of  devoting  a  considerable  amount  of  time  to 


ELECTRICAL  PRINCIPLES  AND  CALCULA  TIONS    101 

mathematical  refinements,  or  of  using  complicated  formulas  of 
which  the  increased  accuracy  is  of  no  practical  value  seeing  that 
they  are  based  upon  assumptions  that  are  never  realized. 

55.  Effect  of  Distributed  Capacity  and  Inductance. — A  long 
transmission  line  of  resistance  R  ohms,  reactance  X  ohms,  and 
capacity  C  farads,  may  be  thought  of  as  consisting  of  n  sections, 
of  resistance  R/n  ohms  and  reactance  X/n  ohms,  with  a  condenser 
of  capacity  C/n  farads  shunting  the  wires  at  the  end  of  each 
section.  The  charging  current  will  fall  off  in  amount  as  the 
distance  from  the  generating  end  increases,  and  the  total  current 
in  the  conductor — being  the  (vectorial)  sum  of  the  charging 
current  and  load  current — will  be  different  in  each  section. 
The  problem  is  further  complicated  by  the  fact  that  the  voltage 
will  not  necessarily  be  the  same  across  all  the  imaginary  con- 
densers, so  that  the  reduction  of  the  charging  current  component 
will  not  even  follow  a  "straight  line"  law.  By  dividing  a  long 
line  into  a  large  number  of  sections,  and  calculating  the  pressure 
drop  and  power  factor  at  the  end  of  each  section,  the  voltage 
drop  and  power  losses  of  the  complete  line  could  be  estimated 
accurately;  but  the  work  would  be  tedious  and,  indeed,  unneces- 
sary. By  imagining  the  number  of  sections,  n,  to  become  larger 
and  larger  without  limit,  we  approach  the  condition  of  distrib- 
uted capacity  for  which  accurate  mathematical  formulas  are 
available.  A  great  deal  of  excellent  work  has  been  done  by  Dr. 
A.  E.  Kennelly,  Dr.  Harold  Pender,  Dr.  J.  A.  Fleming,  Prof. 
H.  B.  Dwight,  and  others1  in  the  matter  of  simplifying  the  exact 
methods  of  calculation  for  long  lines  in  which  the  effects  of  ca- 
pacity are  not  negligible;  but  these  methods  nearly  all  include  the 
use — and  knowledge — of  hyperbolic  functions  in  place  of  the  trigo- 
nometric tables  with  which  all  engineers  are  familiar.  It  is  true 
that  Prof.  Dwight,  following  the  lead  of  Prof.  T.  R.  Rosebrugh, 
has  evolved  a  fairly  simple  means  of  computing  line  voltages2  by 
substituting  the  method  of  convergent  series,  and  using  complex 
quantities,  thus  dispensing  with  the  necessity  for  tables  or  charts 
of  hyperbolic  functions  of  angles;  but,  for  the  solution  of  prac- 
1  Harold  Pender,  Electrical  World  June  8,  1909. 

A.  E.  Kennelly  and  Harold  Pender,  Electrical  World,  Aug.  8,  1914. 

J.  A.  Fleming,  Jour.  Inst.  E.  E.,  p.  717,  Vol.  52,  June  15,  1914. 

A.  E.  Kennelly,  Jour.  Franklin  Inst.,  Sept.,  1914. 

H.  B.  Dwight,  Book  "Transmission  Line  Formulas,"  D.  Van  Nostrand 
Co. 

2  Electrical  World,  Sept.  5,  1914. 

UMNAKY 

--TAW  TVACHn 
MANTA   BARBARA, 


£51 


102  ELECTRIC  POWER  TRANSMISSION 

tical  power  transmission  line  problems,  all  of  these  refined 
methods  of  calculation  are  unnecessary.  From  the  academic 
point  of  view,  the  exact  solution  of  all  engineering  problems  is 
attractive  and  occasionally  desirable,  and  when  the  mathematical 
work  can  be  so  simplified  as  to  be  available  for  use  by  the  average 
engineer,  there  is  no  objection  to  his  using  it.  On  the  other  hand 
if  the  engineer  has  at  his  disposal  simpler  and  shorter  methods  of 
working  which  yield  results  within  the  required  practical  limits  of 
accuracy,  he  should  not  be  criticised  for  preferring  them.  He  may 
base  his  calculations  on  an  assumed  sine  wave  of  e.m.f.,  on  a 
maximum  (sine  wave)  current  of  100  amperes,  and  on  an  esti- 
mated power  factor  of  0.9;  but  he  would  expect  his  line  to 
give  satisfaction  with  an  actual  full  load  current  which  might  be 
anything  between  (say)  90  and  110  amperes,  and  he  would 
consider  himself  lucky  if  the  actual  power  factor  of  the  load  proved 
to  be  within  5  per  cent,  of  the  value  he  had  guessed  at.  It  is 
for  such  reasons  that  the  practicing  engineer — whose  time  is  valua- 
ble, and  who  has  a  habit  of  using  factors  of  safety  somewhat 
freely — rarely  evinces  a  fervent  interest  in  mathematical  re- 
finements whereby  the  (theoretical)  accuracy  of  his  results  may 
be  increased  by  a  small  fraction  of  1  per  cent. 

On  lines  over  50  miles  in  length,  the  effects  of  capacity  on 
voltage  regulation  may  be  appreciable,  and  some  practical 
methods  of  taking  into  account  the  capacity  current  on  long 
transmission  lines  will  be  explained  in  the  following  article. 

66.  Electrical  Calculation  of  Lines  with  Appreciable  Capacity. — 
In  Article  10  of  Chapter  II,  an  example  was  worked  out  showing 
how  the  charging  current  at  the  supply  end  of  a  long  transmission 
line  could  be  calculated,  and  the  effect  of  this  current  in  modi- 
fying the  fundamental  vector  diagram  was  illustrated  in  Fig. 
13.  Thus,  when  we  imagine  a  condenser  to  be  connected  across 
the  lines  at  a  point  where  the  load  current  is  /  amperes  on  a  power 
factor  cos  6,  the  conditions  on  the  supply  side  of  the  condenser 
will  be  a  current  of  /'  amperes  and  a  power  factor  of  cos  6',  these 
modified  values  being  calculated  as  follows. 

The  vector  diagram  Fig.  30  is  the  same  as  Fig.  13  except  for 
a  few  additions.  The  charging  current  can  be  calculated  by 
means  of  the  formula  (36),  which  is 

Ic  =  2irfEnC  (36) 

This  component  of  the  total  line  current  is  drawn  90°  in  advance 


ELECTRICAL  PRINCIPLES  AND  CALCULA  TIONS   103 

of  the  vector  En  =  OB  which  is  the  pressure  across  the  condenser 
terminals.  By  dropping  the  perpendicular  AN  on  OB  and 
making  AM  =  Ic,  we  obtain  OM  =  I'  the  resultant  or  total 
current  in  the  line  on  the  supply  side  of  the  condenser. 

In  order  to  calculate  /'  and  the  new  power  factor  angle  0', 
we  may  write, 

ON=  I  cos6 
NA=  I  sin  6 
NM=  I  sin  e  -  Ic 

NM  7   Sil1    *    ~    !° 


tan  6  -      Ic 


I  cos  e 


FIG.  30. 


whence  0'  and  the  other  trigonometrical  functions  such  as  sin  0' 
and  cos  6'  can  be  read  off  the  slide  rule  or  obtained  from  tables. 
The  line  current  is 


cos  e' 

Using  this  value  of  current,  the  sides  I'R  and  I'X  of  the  line 
impedance  triangle  can  now  be  calculated,  and  the  procedure  for 
calculating  the  line  losses  and  line  drop  will  be  as  described  in 
Article  9  (where  the  line  was  assumed  to  be  without  capacity) 
except  that  /'  and  0'  must  be  substituted  for  I  and  0  in  the 
formulas. 


104  ELECTRIC  POWER  TRANSMISSION 

It  will  perhaps  simplify  matters  and  prevent  confusion  if  some 
of  these  formulas  are  reproduced  here  with  the  necessary  changes 
to  make  them  applicable  to  Fig.  30. 

The  functions  of  the  angle  <£  are 


V  n 
PR  +  En  COS 

^     n 


(42) 

,  ... 

(43) 

I'X  +  En  sin  6' 

tan  *  '  /'B  +  J.^.f  (44) 

From  this  last  formula  (44)  we  obtain  tf>  and  therefore  cos  0 
(the  power  factor  at  the  sending  end  of  the  line),  whence  Vn  of 
which  the  value,  from  (43),  is 


COS  0 

57.  Numerical  Examples  Illustrating  Use  of  Formulas  for  the 
Calculation  of  Power  Factor  and  Voltage  Drop.  —  The  data  for 
use  in  the  calculations  is  as  foilows: 

System;  three-phase. 

Line  pressure  at  receiving  end,  E  =  66,000  volts. 

"Star"  voltage,  En  =  *^™=  33,100  volts. 
v  3 

Frequency,  /  =  60. 

Load  =  3000  k.v.a. 

Power  factor  of  load,  cos  0  =  0.9. 

Length  of  line,  L  =  100  miles. 

Conductors  of  No.  3  copper  cable  (radius  r  =  0.13). 

Spacing,  8  feet  (d  =  96). 

From  wire  table,  the  resistance  is  found  to  be  R  =  1.04  ohms  per 
mile.     By  formula  (29)  the  reactance  per  wire  is 


X  =  0.00466  X  60  X  log  (  1.285  X  ^^ 

=  0.832  ohms  per  mile. 
By  formula  (12)  the  capacity  (wire  to  neutral)  is 

0  0388 
Cm  =  —  —j-  =  0.0135  microfarads  per  mile. 

r 

The  load  current  is  7  =      ^000'00°       =  26.2  amperes. 
\/3  X  66,000 


ELECTRICAL  PRINCIPLES  AND  CALC  ULA  TIONS    105 

By  way  of  illustration,  we  shall  calculate  the  line  drop  by 
imagining  (1)  the  whole  of  the  line  capacity  to  be  concentrated 
at  the  center  of  the  line,  and  (2)  one-half  of  the  total  capacity 
to  be  concentrated  at  each  end  of  the  line. 
Case  (1),  capacity  concentrated  half   way    between   sending    and 

receiving  ends  (Fig.  31). 

With  the  star  voltage  En  =  38,100  at  the  receiving  end,  we 
will  first  calculate  the  voltage  E'n  at  the  point  where  the  con- 
denser is  supposed  to  be  connected.  We  could  if  desired  use  one  of 
the  charts,  Fig.  25  or  27;  but  it  may  be  advisable  to  use,  through- 
out these  examples,  the  trigonometrical  formulas  which  were 
developed  from  the  fundamental  vector  diagram. 

By  formula  (9), 

(26.2  X  50  X  0.832)  +  (38,300  X  0.436) 
(26.2  X  50  X  1.04)  +  (38,100  X  0.9) 


whence  cos  <£  =  0.896  and  sin  <j>  =  0.444. 


1 

/'  Amps 

•  >• 

a 

I     Amps 

t   I  Load  of 
1  Power 
E  1  Factor 
1    I    co,0 

i 

L 

i 

L 

! 

2 

! 

2 

1 

FIG.  31.  —  Diagram  showing  total  capacity  concentrated  at  center  of  line. 

By  formula  (10), 


For  the  other  half  of  the  line  we  must  use  the  formulas  de- 
veloped from  Fig.  30.  By  formula  (36),  the  charging  current 
is 

L  =  27r  X  60  X  39,800  (100  X  0.0135  X  10~6) 

=  20.25  amperes. 
By  formula  (40), 

_  (26.2  X  0.444)  -  20.25 

26.2  X  0.896 
=  -  0.367 

From  tables,  0'  =  20°  9',  cos  0'  =  0.939  and  sin  0'  =  -  0.344 
whence,  by  (41),  the  line  current  in  this  section  is 

/'  -  26.2  =  25  amperes, 


106  ELECTRIC  POWER  TRANSMISSION 

which  is  in  advance  of  the  pressure  across  the  condenser  because 
of  the  negative  sign  resulting  from  the  solution  of  formula  (40). 
The  sides  of  the  impedance  triangle  BCD  (Fig.  30)  are 

I'R  =  25  X  50  X  1.04  =  1300  volts 
and 

I'X  =  25  X  50  X  0.832  =  1040  volts. 

The  procedure  is  now  as  for  the  50  miles  of  line  already  cal- 
culated. Putting  <£'  for  the  power  factor  angle  at  the  sending  end 
of  the  line  (the  angle  0'  =  20°  9'  being  the  power  factor  angle 
at  the  other  end  of  this  section),  we  have  by  formula  (9)  or  (44) 

1040  -  (39,800  X  0.344) 


1 

1300  +  (39, 
-  12,660 

800  X  0.939) 

38,680 

r 

VT^ 

f 

Current  -J 

t|    I    | 

G3 

Load  of 
Power 
Factor 
cot  $ 

BesiBtance   R    ohms:   Beactance    X   ohms 

FIG.  32.  —  Diagram  showing  one  half  of  total  capacity  concentrated  at  each  end 
of  the  line. 

whence  $'  =  18  degrees;  cos  <£'  =  0.951;  and  sin  <f>'  =  —  0.309; 
the  fact  of  sin  0'  being  negative  indicating  a  leading  current. 
By  formula  (10)  or  (45), 


Vn  =  =  40,700  volts. 


The  pressure  drop  per  phase  is  40,700  -  38,100  =  2600  volts, 
or  6.82  per  cent,  of  the  receiving  end  pressure. 

The  current  entering  the  line  at  the  generator  end  under  full 
load  conditions  is  I'  =  25  amperes,  on  a  leading  power  factor 
(cos  <£')  of  0.951. 

It  should  be  mentioned  that  no  high  degree  of  accuracy  is 
claimed  for  these  or  any  numerical  examples  worked  out  in  this 
book.     A  10-inch  slide  rule  is  used  for  all  calculations. 
Case   (2),  half  the  total  capacity  concentrated  at  each  end  of  the 

line  (Fig.  32). 

The  effect  of  connecting  a  condenser  of  capacity  C/2  between 
each  wire  and  neutral  at  the  receiving  end,  is  to  modify  the  power 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS    107 

factor  of  the  load,  the  line  calculations  being  based  on  an  imagi- 
nary load  power  factor  cos  6'  instead  of  the  actual  power  factor 
cos  6.  The  condensers  at  the  sending  end  have  obviously  no 
effect  on  the  line  drop  or  line  losses,  but  they  will  modify  the 
power  factor  of  the  load  at  generator  terminals. 

The  calculations  of  line  drop  are  exactly  as  carried  out  for  the 
supply  end  of  the  line  in  Case  (1).  The  charging  current  is  now 
about  half  the  value  previously  calculated, 

Ic  =  2T  X  60  X  38,100  (50  X  0.0135  X  10~6)  =  9.7  amperes. 
Referring  to  Fig.  30,  we  have,  by  formula  (40), 

(26.2  X  0.436)  -  9.7          „„ 
26.2  X  0.9  =  °-°73 

whence  0'  =  4°  10';  cos  0'  =  0.997,  and  sin  6'  =  0.073. 

0  9 

By   formula    (41)    the   line  current  is  26.2  X  n  'Q7  =  23.65 

u.  v/t/i 

amperes. 

The  resistance  drop  is  I'R  =  23.65  X  100  X  1.04  =  2460  ohms. 
The  reactance  drop  is  I'X  =  23.65  X  100  X  0  832  =  1970  ohms. 
By  formula  (44)  , 

1970  +  (38,100  X  0.073) 
tan  *  =  2460  +  (38,100  X  0.997)  = 

whence  cos  0  =  0.994,  and  sin  <£  =  0.110. 
Thus,  by  formula  (45), 


-  «,700  volts 

which  is  the  same  as  the  figure  obtained  by  assuming  the  whole 
of  the  capacity  to  be  concentrated  at  the  center  of  the  line. 

If  it  is  desired  to  calculate  the  current  and  power  factor  at  the 
generator  terminals,  we  have  merely  to  repeat  the  process  by 
putting  the  new  numerical  values  in  the  formulas.  Thus 

407 

Ic  =  9.7  1^-  =  10.35  amperes, 
ool 

and  the  formula  (40)  when  re-written,  becomes 
/'  sin  <t>  -  Ic 


tan  <£'  = 


/'  cos  (f> 
(23.65  X  0.110)  -  10.35 
23.65  X  0.994 


108  ELECTRIC  POWER  TRANSMISSION 

whence  cos  0'  =  0.95  (leading),  and  by  formula  (41),  the  line 
current  is 


- 
23.65         p  =  24.8  amperes. 


In  this  example,  the  capacity  current  is  relatively  large  because 
the  maximum  load  on  a  line  100  miles  long  would  generally  be 
more  than  the  assumed  value  of  3000  k.v.a.;  but  nevertheless, 
both  methods  give  approximately  the  same  results.  Comparing 
the  figures,  and  bearing  in  mind  that  no  high  degree  of  accuracy 
is  claimed,  we  have, 


Approximation  (1) 
(Fig.  31) 

Approximation  (2) 
(Fig.  32) 

Percentage  line  drop  
Line  current  at  generating  end  
Power  factor  at  generating  end  .... 

6.82 
25  .  00  amperes 
0.951  (leading) 

6.82 
24.80  amperes 
0.95  (leading) 

Either  method  gives  results  that  are  sufficiently  accurate  for 
practical  purposes,  even  in  the  case  of  long  high-voltage  lines.1 

58.  Distinction  Between  Regulation  and  Line  Drop. — The 
percentage  drop  of  pressure  on  a  transmission  line  may  be  de- 
fined as  the  difference  between  the  sending  end  and  receiving 
end  pressures  expressed  as  a  percentage  of  the  receiving  end 
pressure.  Thus, 

V  —  E 

Per  cent,  pressure  drop  =  — ^ —  X  100 
Hi 

and  this  is  the  same  as  the  regulation  when  the  capacity  current 
is  so  small  as  to  be  negligible.  When  the  capacity  current  is 
appreciable,  it  will  cause  the  voltage  of  the  unloaded  line  to  be 
greater  at  the  receiving  than  at  the  generating  end,  as  explained 
in  Article  10,  Chapter  II  (Fig.  11);  and  since  the  regulation  is 
defined  as  the  change  of  pressure  at  the  receiving  end  when  the 
load  is  thrown  off  (the  supply  voltage  remaining  constant),  the 
regulation  of  a  long  high-voltage  transmission  line  will  usually 
be  greater  than  the  pressure  drop. 

1  The  manner  in  which  the  total  current  in  a  long  line  of  appreciable 
capacity  changes  both  in  magnitude  and  phase,  may  be  illustrated  graphic- 
ally by  means  of  models  or  diagrams  involving  the  idea  of  two  planes  per- 
pendicular to  each  other.  The  writer  has  in  mind  diagrams  similar  to  those 
used  by  Prof.  D.  D.  Ewing  in  the  Electrical  World  of  Dec.  29,  1917,  Vol.  70, 
p.  1252. 


ELECTRICAL  PRINCIPLES  AND  CALCULA  TIONS    109 

The  rise  of  pressure  at  the  end  of  a  long  transmission  line  is 
independent  of  the  size  and  spacing  of  the  wires.  It  may  be 
calculated  approximately  as  follows. 

In  Fig.  33,  the  IR  drop  (CB)  due  to  the  charging  current  Ic, 
may  be  neglected  as  it  has  no  appreciable  effect  on  the  pressure 
rise  (En  —  Vn)  which  we  shall  therefore  consider  as  being  equal 
to  the  induced  volts  DC.  By  formula  (36)  the  charging  current 
is 

Ic  =  27r/#nCfmL  X  10-6 

where  Cm  is  the  capacity  in  microfarads  per  mile,  and  L  is  the 
length  of  the  line  in  miles.     The  induced  volts  are, 

IX  =  2irfLIcL 


Uc) 

(Vnl 

Fio.  33. — Vector  diagram  showing  pressure  rise  at  end  of  long  unloaded  line. 

which,  after  substituting  the  above  value  of  Ic,  become 

IX  =  (2*f)zEn(CmL)L*  X  10-6 

but,  by  formula  (37),  the  product  CmL  has  the  constant  value 
04  TYjTy  whence  the  pressure  rise  (conductor  to  neutral)  of  the 
unloaded  line  is 

IX  =  (En  -  Vn)  =  (2irf)zEnL2  X  2.88  X  10-11 

=  l.!4Enf*L*  X  10-9  volts  (46) 

which  assumes  sinusoidal  wave  shapes. 

This  pressure  rise,  expressed  as  a  percentage  of  the  line  voltage 
is 

Per  cent,  pressure  rise 

due  to  capacity  and  [  =  1.14/2!/2  X  10~7  (47) 

inductance  of  line 


110  ELECTRIC  POWER  TRANSMISSION 

The  regulation,  on  the  sine  wave  assumption,  is  therefore 
equal  to  Percentage  line  drop  +  l.l^L2  X  10~7;  but  the  last 
term  is  negligible  unless  the  distance  of  transmission  (L)  is  great. 

As  an  example  of  the  application  of  formula  (47),  the  per- 
centage line  drop  in  the  numerical  problem  of  the  last  article 
was  6.82  and  the  percentage  rise  at  the  end  of  the  unloaded  line 
is  approximately  1.14  X  (60)2  X  (100)2  X  10~7  =  4.11,  whence 
the  regulation  is  6.82  +  4.11  =  10.93  per  cent. 

59.  Line  Losses. — Apart  from  leakage  and  corona  losses, 
which  will  be  considered  in  Chapter  V,  the  watts  lost  in  transmis- 
sion are  the  I2R  losses.  On  a  three-phase  line  these  will  be 

w  =  3PRL 

where  R  is  the  resistance  per  mile  of  conductor — corrected  if 
necessary  for  skin  effect — and  L  is  the  distance  of  transmission, 
in  miles. 

When  the  current  entering  the  line  at  the  sending  end  is  equal 
to  the  current  leaving  the  line  at  the  receiving  end,  the  value  to 
take  for  /  in  the  above  formula  is  simply  the  load  current;  but 
when  the  effect  of  the  distributed  capacity  becomes  important 
— as  on  a  long-distance  high-voltage  transmission — the  question 
arises  as  to  what  particular  value  of  the  current  should  be  used 
in  the  calculations  for  line  loss. 

As  an  alternative  to  dividing  the  line  into  a  large  number  of 
sections  and  calculating  the  current  in  each  section — which 
would  involve  a  considerable  amount  of  tedious  work — we  can 
calculate  the  average  value  of  the  square  of  the  current  over  the 
whole  distance  of  transmission.  This  calculation  is  easily  made 
if  the  effect  of  voltage  drop  is  neglected,  or,  in  other  words,  if 
the  amount  of  the  charging  current  is  supposed  to  diminish  in 
direct  proportion  to  the  distance  from  the  supply  end  of  the  line. 
Thus,  if  Ic  is  the  value  of  the  charging  current  at  the  sending 
end,  the  value  of  the  charging  current  component  of  the  total 
current  at  a  point  x  miles  from  the  generating  end  of  a  line  L 
miles  long  will  be 

Ix  =  ^  (L  -  x)  (48) 

Referring,  now,  to  the  vector  diagram  Fig.  34,  where  7  is  the 
load  current  and  cos  6  is  the  power  factor  of  the  load,  the  total 
line  current  (7j)  at  a  point  x  miles  from  the  generating  end  is 
the  sum,  or  resultant,  of  7  and  Ix.  This  resultant  can  be  ex- 


ELECTRICAL  PRINCIPLES  AND  CALC  ULA  TIONS    1 1 1 


pressed  in  terms  of  its  "in-phase"  and  "wattless"  components 
thus: 

Ii  =  V(o  -  /z)2  +  62 

and  the  average  value  of  the  square  of  this  quantity  is  62  +  average 
value  of  (a  —  7Z)2  as  x  increases  from  zero  to  its  maximum  value  L. 
Average  of 


b  =/COB    0 


a=Isin  0 


FIG.  34. — Vector  diagram  showing  total  current  at  any  distance  from  end  of  line. 

Adding  62  to  this  quantity,  we  get  for  the  average  value  of  the 
square  of  the  line  current 


(/*2)average  =    (&2  +  «2)    ~   «/,   +  |/c2 

=  P  -  ICI  sin  e  +  ^/t2 


(49) 


and  the  watts  lost  in  a  three-phase  line,  neglecting  corona  and 
leakage  losses,  are  approximately 

W  =  3RL    (/^average  (50) 

If  the  current  waves  are  not  sinusoidal,  and  if  the  pressure  at 
the  sending  end  is  appreciably  higher  than  the  receiving  end 
pressure,  the  average  square  of  the  current  will  not  be  quite 
correctly  obtained  from  formula  (49). 


112  ELECTRIC  POWER  TRANSMISSION 

Example  ofLineLoss  Calculations. — Instead  of  assuming  entirely 
new  conditions,  we  shall  use  the  data  of  the  numerical  example 
in  Article  10  of  Chapter  II.  and  calculate  the  line  losses  (1) 
when  there  is  no  load  at  the  receiving  end,  and  (2)  when  the 
load  at  the  receiving  end  is  10,000  k.w.  with  a  power  factor  of 
0.9. 

Assuming  the  conductors  to  be  No.  00  copper  throughout,  the 
resistance  per  mile  of  wire  (neglecting  skin  effect)  will  be  0.41 
ohm,  and  the  known  quantities  are  therefore: 

Load  at  receiving  end  =  \/3  El  cos  6  =  10,000,000  watts. 

Line  voltage,  E  =  100,000. 

Load  power  factor,  cos  0  =  0.9. 

Load  current,  /  =  64.2  amperes. 

Resistance  of  conductors,  R  =  0.41  ohms  per  mile. 

Distance  of  transmission,  L  =  210  miles. 

Capacity  current  at  generating  end  "1 

J  .  .  .,  >     =  63. 6  amperes, 

(as  previously  calculated)  J 

(1)  By  formula  (49)  the  average  value  of  the  square  of  the 
charging  current  when  I  =  0  is 

(/coverage  =  j^2  =    1345. 

By  formula  (50)  the  total  line  loss  is  3  X  1345  X  0.41  X  210 
X  10~3  =  348  k.w.  This  is  the  true  output  of  the  generating 
station  (neglecting  corona  and  leakage  losses)  when  the  working 
voltage  is  applied  to  the  unloaded  line;  but  the  k.v.a.  or  apparent 

kilowatt  output  is  -\/3  X  100,000  X  63.6  X  10~3  =  11,000  k.v.a. 

(2)  By  formula  (49)  the  average  value  of  the  square  of  the  line 
current  when  the  load  current  is  64.2  amperes  on  a  power  factor 
cos  6  =  0.9  (sin  6  =  0.436)  will  be 

(/^average  =  (64.2)2  X  1345  -  (63.6  X  64.2  X  0.436)  =  3680 
whence  the  total  line  loss  at  full  load  is  3  X  3680  X  0.41  X 
210  X  10-3  =  952  k.w. 

60.  Control  of  Voltage  on  Transmission  Lines. — The  pressure 
drop  at  the  receiving  end  of  a  transmission  line  may  be  compen- 
sated for  by  raising  the  voltage  at  the  generating  end  as  the  load 
increases.  There  are  obvious  disadvantages  to  such  a  method  of 
operation,  and  it  is  better,  if  possible,  to  regulate  the  voltage  at 
the  point,  or  points,  where  constant  pressure  is  required.  This  is 


ELECTRICAL  PRINCIPLES  AND  CALC ULA  TIONS    1 13 

especially  true  of  transmission  lines  on  which  there  are  substa- 
tions or  branch  lines  at  intermediate  points. 

The  necessary  steady  voltage  at  the  receiving  points  may  be 
obtained  by  installing  hand  operated  or  automatically  controlled 
variable-ratio  transformers  or  "boosters,"  which  may  be  either 


£      S 

FIG.  35. — "Boosters"  on  three-phase  system. 

of  the  type  with  movable  iron  core,  or  with  tappings  from  the 
windings  taken  out  to  a  multiple-contact  regulating  switch. 

On  a  delta-connected  three-phase  system  it  is  not  necessary  to 
provide  more  than  two  single-phase  regulators  as  these  may  be 
connected  up  as  indicated  in  Fig.  35.  Here  P  and  S  represent 


B' 

FIG.  36. — Vector  diagram — two  "boosters"  on  three-phase  system. 

respectively  the  primary  and  secondary  windings  of  the  variable- 
ratio  transformers.  That  these  are  capable  of  raising  the  voltage 
equally  on  all  three  phases  to  the  extent  of  the  volts  induced  in 
the  secondary  coil  S  will  be  clear  from  an  inspection  of  Fig.  36. 
In  this  diagram,  AB,  BC,  and  CA  are  the  three  vectors  repre- 
senting the  pressures  e  before  boosting  up:  A  A'  and  BB'  repre- 


114  ELECTRIC  POWER  TRANSMISSION 

sent  the  added  volts  between  the  terminals  A  and  A'  or  B  and 
B'  (Fig.  35).  These  added  volts  are  evidently  in  phase  with 
the  pressures  indicated  by  the  vectors  CA  and  BC  respectively, 
because  the  potential  difference  at  the  secondary  terminals  of  a 
well-designed  transformer  is  always  in  phase  with  the  primary 
impressed  e.m.f.  It  is  only  necessary  to  complete  the  triangle 
CA'B'  to  see  that  the  two  transformers  connected  up  in  the 
manner  described  will  do  all  that  is  required  in  the  way  of  raising 
the  pressure  on  the  three-phase  circuit. 

61.  Effect  of  Boosting  Voltage  at  Intervals  Along  a  Trans- 
mission Line. — If  a  long  transmission  line,  insulated  for  a  maxi- 
mum working  pressure  of  (say)  100,000  volts,  can  be  worked  as  a 
100,000-volt  line  at  all  times  through  its  entire  length,  it  will  be 
more  efficient  than  if  only  a  portion  of  it  is  working  as  a  100,000- 
volt  transmission  while  portions  farther  from  the  generating  end 


100  Amps. 


20.000 


,000 


\Loss  in  1st  Section  =  500  KW.  X  Loss  in  2nd  Section  =  500  KW. 

FIG.  37.  —  Method  of  maintaining  pressure  on  long  line. 

are  working  at  (say)  80,000  volts.  By  installing  boosters  along 
the  line  to  maintain  the  pressure  at  or  near  the  maximum  working 
value,  whatever  the  load  may  be,  economies  may  frequently  be 
effected.  It  is  true  that  the  energy  put  into  the  line  at  inter- 
mediate points  cannot  be  cheaper  —  and,  indeed,  is  usually  more 
costly  —  than  the  energy  supplied  to  the  line  at  the  generating 
end;  but  the  booster  system  allows  of  the  pressure  being  kept 
up  all  along  the  line,  thus  effecting  economy;  provided  always 
that  the  losses  in  the  boosters  themselves,  their  maintenance, 
and  the  necessary  allowances  for  interest  and  depreciation,  do 
not  counterbalance  the  saving. 

As  an  example,  consider  a  single-phase  line  conveying  a  current 
of  100  amperes  at  an  initial  pressure  of  20,000  volts.  Suppose  the 
drop  in  pressure  in  the  whole  length  of  line  to  be  as  great  as  10,000 
volts;  this  will  leave  only  10,000  volts  at  the  receiving  end. 

The  power  put  into  the  line  at  generating  end  =  2000  k.w. 

The  loss  hi  the  line  =  1000  k.w. 

The  power  available  at  receiving  end  =  1000  k.w. 

Hence,  efficiency  of  line  =  50  per  cent. 


ELECTRICAL  PRINCIPLES  AND  CALCULA  TIONS   115 

Now  imagine  a  booster  to  be  introduced  at  a  point  half-way 
along  the  line.  This  booster  may  be  considered  as  a  suitably 
insulated  alternator  of  500  k.w.  capacity,  capable  of  generating 
100  amperes  at  5000  volts,  the  arrangement  being  as  shown  in  Fig. 
37.  The  drop  in  the  first  section  of  the  line  is,  as  before,  5000 
volts;  and  the  drop  in  the  second  section  is  evidently  similar — 
namely,  5000  volts — which  means  that  the  total  amount  of  power 
dissipated  in  the  line  is  the  same  as  it  was  before  the  booster  was 
introduced.  But  by  providing  this  booster  at  the  middle  point 
of  the  line,  it  has  been  possible  to  raise  the  pressure  at  this  point 
up  to  the  initial  value  of  20,000  volts,  with  the  result  that  15,000 
volts  (1500  k.w.)  are  available  at  the  receiving  end.  The  addi- 
tional power  available  for  useful  purposes  has,  of  course,  cost 
something  to  produce;  but  the  point  to  be  noted  is  this:  by  keep- 
ing up  the  pressure,  it  has  been  possible  to  transmit  a  greater 


FIG.  38.  —  Transformers  connected  as  "boosters"  on  transmission  line. 

amount  of  energy  to  the  receiving  end  of  the  line  without  increas- 
ing the  losses  in  the  conductors.  If,  for  the  sake  of  simplicity,  the 
losses  in  the  booster  are  neglected,  the  line  efficiency  is  arrived 
at  thus: 

Power  supplied  to  the  line  =  2000  +  500  =  2500  k.w. 

Power  lost  in  the  line  =  1000  k.w. 

Power  available  at  receiving  end  =  1500  k.w. 

Line  efficiency  =  ^r^-:  =  60  per  cent. 


Boosters  may  be  arranged  to  take  their  power  from  the  generat- 
ing end  of  the  line;  that  is  to  say,  they  may  take  the  form  of 
variable-ratio  transformers,  with  hand  or  automatic  regulation, 
connected  up  as  indicated  in  Fig.  38.  Transformers  so  connected 
will  provide  the  additional  volts  at  the  cost  of  a  corresponding 
loss  of  current. 

62.  Control  of  Power  Factor.  —  The  advantages  of  operating 
alternating  machinery  and  systems  on  unity  power  factor,  when 
possible,  are  so  well  known  that  it  will  not  be  necessary  to  dis- 
cuss the  matter  here.  The  line  losses  alone  —  as  shown  in  Article 


116  ELECTRIC  POWER  TRANSMISSION 

6  of  Chapter  II — are  inversely  proportional  to  the  square  of  the 
power  factor.  Thus,  if  the  total  I*R  loss  in  the  line  were  200 
k.w.  with  a  power  factor  of  0.707  (a  by  no  means  impossible 
figure  in  practice),  this  loss  would  be  reduced  to  100  k.w.  if  the 
same  total  power  could  be  transmitted  at  unity  power  factor. 

The  control  of  power  factor  is  obtained  by  balancing  any 
excess  of  inductive  reactance  with  condensive  reactance,  or  vice 
versa.  With  a  changing  load  at  the  end  of  a  long  transmission 
line,  the  power  factor  at  any  given  point  on  the  line  is  continually 
changing,  even  if  the  power  factor  of  the  load  remains  constant; 
and  the  most  convenient  means  of  providing  the  reactance 
necessary  to  maintain  a  constant  and  improved  power  factor 
is  to  install  synchronous  motors  which  can  be  made  to  draw 

Synchronous  Booster  Controlling 
Line  Prewure 


Synchronous  Motor 
Controlling  Power  Factor 

Load  1 


FlG.    39. 

leading  or  lagging  currents  from  the  line  by  over-  or  under- 
exciting  their  field  magnets. 

The  principles  underlying  the  fact  that  an  alternating  current 
synchronous  motor  can  be  made  to  act  either  as  a  rotary  con- 
denser or  a  rotary  reactor  have  been  dealt  with  by  the  writer 
elsewhere,1  and  they  are,  moreover,  so  generally  understood  that 
we  shall  assume  the  power  factor  to  be  capable  of  control  by 
merely  providing  one  or  more  synchronous  motors  of  sufficient 
capacity  and  with  the  necessary  equipment  for  varying  the  field 
current,  at  the  desired  point  on  the  transmission  line. 

Bearing  in  mind  that  power  factor  control  is  quite  as  impor- 
tant, if  not  more  important,  than  voltage  control  so  far  as  the 
losses  and  efficiency  of  transmission  are  concerned,  the  arrange- 
ment shown  diagrammatically  in  Fig.  39  should  be  preferable  to 
the  arrangements  of  Figs.  37  and  38. 

Here  B  is  a  synchronous  generator  connected  as  a  "booster" 
and  provided  with  field  regulation  in  order  to  control  the  voltage. 
It  is  direct-coupled  to  the  synchronous  motor,  M,  which  is  pro- 
vided with  field  regulation  in  order  to  control  the  power  factor. 

1  Polyphase  Currents,  Whittaker  and  Co.,  London. 


ELECTRICAL  PRINCIPLES  AND  CALC  ULA  TIONS   117 

Before  deciding  to  install  synchronous  machinery  to  control 
the  voltage  and  power  factor  of  a  transmission  line,  it  is  neces- 
sary to  consider  the  increased  cost  due  to  such  machinery  over 
alternative  methods  of  pressure  control,  and  compare  this  with 
the  capitalized  cost  of  the  saving  in  transmission  losses  due  to 
the  improved  power  factor.  A  point  which  should  not  be  over- 
looked is  the  possibility  of  synchronous  machinery  falling  out 
of  step,  and  so  causing  troubles  and  interruptions  to  supply  which 
are  best  avoided  by  installing  no  auxiliary  apparatus  which  it 
is  possible  to  do  without. 

63.  Use  of  Rotary  Reactors  to  Control  the  Voltage. — Although 
Fig.  39  shows  two  synchronous  machines,  each  with  a  particular 
function  to  perform,  it  is  not  always  necessary  to  maintain  the 
power  factor  at  a  constant  value,  and  if  the  machine  M  is  of 


FIG.  40. — Vector  diagram  showing  components  of  voltage  which  cause  pressure 
drop. 

sufficient  capacity  and  properly  designed,  it  may  be  used  to  main- 
tain constant  voltage  without  the  addition  of  the  second  machine 
(B)  connected  in  series  with  the  line.  The  reason  for  this  is  that 
by  changing  the  power  factor — or  the  phase  difference  between 
the  line  current  and  e.m.f. — the  inductance  of  the  line  and  of 
such  apparatus  as  transformers  connected  thereto,  can  be  util- 
ized to  provide  the  required  voltage.  This  is  best  explained 
with  the  aid  of  vector  diagrams. 

Fig.  40  is  similar  to  the  fundamental  regulation  diagrams 
(Figs.  10  and  26)  except  that  the  ohmic  and  reactive  voltage 
components  of  the  total  pressure  drop  are  shown  separately  for 
the  "in-phase"  and  "wattless"  components  of  the  total  line 
current.  Thus,  if  En,  I,  and  cos  6,  stand  respectively  for  the 


118 


ELECTRIC  POWER  TRANSMISSION 


"star"  voltage,  the  line  current,  and  the  power  factor  at  the  re- 
ceiving end,  the  IR  and  IX  drops  due  to  the  "in-phase"  com- 
ponent, ON,  of  the  current  are  CB  and  HC  respectively,  while 
the  IR  and  IX  drops  due  to  the  "wattless"  component,  NA,  of 
the  current  are  GH  and  DG  respectively.  The  pressure  neces- 
sary at  the  generating  end  is  FB  =  OD,  and  the  pressure  drop 
is  DF  =  Vn  —  En.  If,  now,  we  can — by  means  of  overexcited  syn- 
chronous machinery — so  change  the  power  factor  that  the  point 
D  will  fall  on  the  dotted  circle  of  radius  OB,  we  shall  obtain  the 
condition  of  constant  voltage,  i.e.,  the  same  voltage  at  the 
generating,  as  at  the  receiving,  end  of  the  line. 


FIG.  41. — Vector  diagram  illustrating  effect  of  reactors  in  maintaining  constant 
voltage. 

Assuming  no  change  in  the  load  current  /,  the  impedance 
triangle  BCH  will  also  remain  unaltered;  but,  by  drawing  a 
leading  "wattless"  current  from  the  line,  the  current  compo- 
nent NA  can  be  not  only  annulled,  but  actually  reversed,  thus 
making  the  line  current  lead  the  receiving  end  voltage,  there- 
by changing  the  voltage  drop  due  to  the  reactive  component  of 
the  current  into  a  voltage  rise.  This  is  shown  in  Fig.  41  where 
the  excitation  of  synchronous  reactors,  connected  across  the 
line  at  the  receiving  end,  has  been  increased  until  the  leading 


ELECTRICAL  PRINCIPLES  AND  CALC ULA  TIONS    1 19 

component  of  the  line  current  is  equal  to  OK.  The  resultant 
current  is  OM  with  a  leading  "wattless"  component  NM,  giv- 
ing the  impedance  triangle  HGD  which  throws  the  point  D  on  the 
dotted  circle  and  makes  Vn  =  En. 

This  is  the  principle  of  constant  voltage  transmission  with  the 
pressure  regulation  obtained  by  providing  the  necessary  number 
of  variable-field  synchronous  motors  at  suitable  points  on  the 
transmission  line,  but  mainly  at  the  receiving  end  where  the 
heavy  load  is  taken  off.  By  this  method,  the  reactance  of  a 
long  line — usually  an  objectionable  feature  tending  to  limit  the 
size  of  individual  conductors — is  actually  necessary  to  the  proper 
regulation  of  the  voltage.  The  machines  used  as  "rotary 
reactors"  may  be  synchronous  motors  from  which  mechanical 
power  is  obtained,  or  rotary  converters,  or  again,  machines 
specially  designed  for  no  other  purpose  than  to  regulate  the 
amount  and  direction  of  the  "wattless"  current,  in  which  case 
they  would  be  installed  in  the  receiving  point  substations  and 
would  be  run  "idle."  The  power  factor  will  not  necessarily  be 
unity,  but  the  improvement  in  the  power  factor  will  generally 
permit  of  more  energy  being  transmitted  along  a  given  line  than 
would  otherwise  be  permissible  or  economical.  The  most  econom- 
ical cross  section  of  conductor  may  be  used  without  regard  to 
pressure  drop,  because  where  a  drop  of  10  to  15  per  cent,  would 
be  about  the  upper  limit  with  the  older  systems  of  regulation,  a 
drop  of  25  per  cent,  (due  mainly  to  the  inductance  of  the  line) 
can  be  taken  care  of  by  synchronous  reactors. 

One  of  the  most  ardent  advocates  of  this  system  of  regulation 
is  Prof.  H.  B.  Dwight,  whose  book1  should  be  consulted  by  those 
desiring  further  information  on  this  subject. 

The  charging  current  of  the  line  was  not  referred  to  in  connec- 
tion with  the  diagram  Fig.  41,  but  it  is  evident  that  it  must  to 
some  extent  be  helpful  in  reducing  the  necessary  size  of  the 
synchronous  motors,  of  which  the  capacity  is  determined  by  the 
amount  of  "wattless"  current  that  they  are  able  to  provide. 

The  regulation  of  power  factor  (and  incidentally  of  the  voltage) 
by  means  of  synchronous  motors  is  not  applicable  to  short-dis- 
tance small-power  transmissions,  and  even  on  long-distance  lines 
transmitting  large  amounts  of  energy,  the  engineer  should  be 
careful  to  consider  the  whole  problem  from  the  economic  point  of 
view:  there  are  a  great  many  factors  to  be  taken  into  account, 

1  "Constant  Voltage  Transmission,"  by  H.  B.  Dwight,  John  Wiley  &  Sons. 


120  ELECTRIC  POWER  TRANSMISSION 

among  which  reliability  of  service  and  maintenance  costs  are  not 
the  least  important. 

The  space  taken  up  by  this  discussion  of  a  particular  system 
of  control  may  seem  excessive  in  view  of  the  limitations  of  this 
book;  but  with  the  improvements  hi  design  of  electrical  machinery 
and  the  increasing  magnitude  of  power  transmission  schemes, 
there  is  a  possibility  of  the  system  being  used  extensively  in  the 
future.  It  is  advocated  not  only  by  the  manufacturers  of  syn- 
chronous alternating-current  machinery,  but  by  engineers  who 
have  satisfied  themselves  that  economy  and  good  service  can, 
under  favorable  conditions,  be  obtained  thereby.  A  notable 
instance  is  the  transmission  line  from  the  hydroelectric  plant  at 
Point  du  Bois  in  Canada  to  the  city  of  Winnipeg  where  two  6000- 
k.v.a.  synchronous  motors  with  automatic  regulation  are  in- 
stalled for  the  sole  purpose  of  regulating  the  voltage  by  power 
factor  control;  the  line  pressure  at  the  generating  station 
remaining  constant. 

64.  Power  Factor  of  Load. — The  power  factor  of  the  load  is 
not  always  easy  to  estimate;  it  may  consist  of  induction  motors 
of  various  sizes,  together  with  lighting  circuits,  all  having  different 
power  factors.  If  several  circuits  of  different  power  factors  are 
connected  in  parallel,  the  joint  power  factor  may  be  calculated 
by  the  formula: 

Cos  e  =  \     .  =  (51) 


J. 

/         //!  sin  0!  +  72  sin  02  +    . 
\      h  Ui  cos  0i  +  72  cos  02  +    . 


where  /i,  72  .  .  .  are  the  currents  taken  by  the  various  circuits 
of  power  factors  cos  0i,  cos  62,  .  .  .  etc. 

The  formula  (51)  is  easily  developed  by  summing  up  the 
"wattless"  and  "in-phase"  components  of  the  various  currents 
separately.  The  quantity  in  brackets  is  thus  seen  to  be  tan  6, 
while  the  complete  formula  is  derived  from  the  well-known 
relation 

1 


cos2  e 


1  +  tan2  6 


65.  Grounded    versus    Isolated    Transmission    Systems. — 

Whether  or  not  it  is  advisable,  on  three-phase  transmissions, 
to  use  the  star  connection  with  grounded  neutral,  or  a  system— 
either  star  or  delta — without  any  connection  to  ground,  is  not  a 
matter  of  very  great  importance;  and  since  no  theoretical  con- 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS    121 

elusions  based  on  general  principles  have  been  arrived  at,  the 
engineer  is  compelled  to  consider  each  particular  case  on  its  own 
merits,  and  be  guided  by  practical  results  obtained  under  similar 
conditions. 

With  a  view  to  eliminating  the  third  harmonic  and  its  multiples, 
and  so  obtaining  as  nearly  as  possible  a  sine  wave  of  e.m.f.,  the 
generators  are  usually  Y  connected,  a  practice  which  has  the 
further  advantage  that  the  neutral  point  can  be  readily  grounded 
if  desired.  The  low-tension  windings  of  the  transformers,  both 
at  generating  and  receiving  ends  of  the  line  are  generally  delta 
connected;  but  so  far  as  the  high-tension  windings  are  concerned, 
these  may  be  star  at  both  ends,  or  delta  at  both  ends,  or  star  at 
one  end  and  delta  at  the  other.  Then  again,  the  neutral  point 
of  a  high-tension  system  may  be  connected  to  ground  either 
directly  or  through  a  resistance,  the  results  being  by  no  means 
the  same  in  the  two  cases. 

The  object  of  grounding  the  neutral  of  a  high-tension  system 
is  mainly  to  protect  the  insulation  from  abnormally  high  pres- 
sures which  might  aggravate  the  trouble  in  the  event  of  a  ground 
occurring  on  one  wire,  and  so  lead  to  serious  interruption  of 
service.  It  is,  in  fact,  the  question  of  line  insulation  considered 
in  connection  with  continuity  of  service  which  is  generally  the 
determining  factor  in  deciding  whether  or  not  the  neutral  shall 
be  grounded,  and  whether  the  grounding,  if  adopted,  shall  be 
with  or  without  the  intervention  of  a  resistance. 

Other  considerations,  such  as  the  effect  on  neighboring  tele- 
phone lines,  both  under  normal  conditions  of  working  and  when  a 
breakdown  occurs,  will  also  influence  the  decision;  and  it  is 
hardly  possible  in  this  place  to  add  much  to  what  has  already 
been  said  in  Article  7  of  Chapter  II;  there  is  no  general  rule  to  be 
followed  seeing  that  the  circumstances  of  each  individual  case 
will  have  a  bearing  on  the  settlement  of  this  question. 

There  would  appear  to  be  no  particular  object  in  grounding 
well-insulated  systems  of  moderate  pressures — up  to,  say,  60,000 
volts;  but,  in  the  large  high-pressure  transmissions,  especially 
with  an  extended  system  of  branch  circuits  and  tie  lines,  a  dead- 
grounded  neutral  is  usually  desirable.  Any  three-phase  trans- 
mission system  on  which  the  insulation  is  likely  to  give  trouble 
when  the  pressure  to  ground  is  raised  in  the  proportion  of  \/3  : 1 
should  have  the  neutral  grounded,  either  directly  or  through  a 
suitably  proportioned  resistance. 


122  ELECTRIC  POWER  TRANSMISSION 

66.  Interference  Between  Power  and  Telephone  Lines. — Al- 
though the  question  of  interference  between  power  lines  and 
neighboring  telephone  lines  is  a  very  important  one,  it  is  also  a 
difficult  one  to  settle  satisfactorily  or  even  discuss  adequately 
in  a  book  dealing  primarily  with  the  design  of  high  tension  trans- 
mission lines.     The  problem  is  of  particular  interest  to  the  tele- 
phone engineer  who  will  no  doubt  ultimately  find  a  satisfactory 
remedy  for  the  very  real  troubles  which  are  liable  to  occur — 
especially  when  abnormal  conditions  lead  to  unbalancing  of  the 
power  load — when  telephone  wires  run  parallel  to  alternating- 
current  power  lines  for  a  considerable  distance. 

It  is  a  comparatively  easy  matter  to  calculate  the  flux  of 
induction  which  the  current  in  the  power  conductors  will  set  up 
in  the  loop  formed  by  the  telephone  wires,  and  by  carefully 
planned  and  frequent  transpositions,  this  effect  can  be  greatly 
reduced  if  not  entirely  overcome;  but  the  electrostatic  effects  are 
probably  of  greater  importance  because  they  are  less  easily  dealt 
with. 

67.  Insulation  of  Telephone  Lines. — It  has  only  lately  been 
realized  that  one  of  the  essential  requirements  for  telephone  lines 
strung  on  the  same  supports  as,  or  very  close  to,  high-pressure 
power  conductors,  is  high  insulation.     This  good  insulation  is 
necessary  to  prevent  puncture  of  the  insulators  when  high  poten- 
tials are  induced  on  the  telephone  wires  at  times  of  abnormal 
conditions — such   as   intermittent   short    circuits,    or   lightning 
disturbances — on   the   power   lines.     As   an   example   of  good 
practice  in  this  respect,  the  Georgia  Railway  &  Power  Co.  have 
provided  insulators  suitable  for  a  working  pressure  of  22,000  volts 
to  carry  the  telephone  wires  which  parallel  their  110,000- volt 
power  lines  from  Atlanta  to  Tellulah,  Ga. 

68.  Electrostatic  Induction. — The  dielectric  field  due  to  the 
alternating  voltage  of  the  power  conductors  induces  a  varying 
charge  upon  the  neighboring  telephone  wires.     If  each  wire  of 
the  telephone  line  were  at  an  equal  average  distance  from  each 
conductor  of  the  power  line,  there  would  be  no  difference  of  poten- 
tial created  between  the  two  sides  of  the  telephone  receiver,  and 
there  should  be  no  buzzing,  etc.,  due  to  this  cause.     In  other 
words,  with  adequate,  properly  worked  out  transpositions,  the 
capacity  currents  passing  between  the  power  line  and  the  tele- 
phone line  will  not  pass  through  the  telephone  receiver.     But, 
even  if  the  electrostatic  flux  is  at  all  times  of  the  same  kind  and 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS    123 


amount  for  both  wires  of  a  telephone  circuit,  this  does  not  prevent 

the  telephone  circuit  as  a  whole  being  subject  to  alternating 

pressures  relatively  to  ground,  and  these  pressures  may  reach  high 

values,  depending  upon  the  voltage  of  the  power  line  and  the  prox- 

imity of  the  two  (parallel)  circuits.     If  the  telephone  circuit  is 

grounded,  the  charging  current  passing  between  the  power  con- 

ductors and  the  telephone  line  will  find  its  way  to  ground  by 

flowing    along   the    telephone   wires, 

and  since  this  current  may  amount 

to  several  amperes,  trouble  is  almost 

certain    to    occur    unless    what    are 

known   as  "drainage   coils"  are  pro- 

vided.    If  a  choke  coil  with  an  iron 

core  —  such  as  the  primary  of  an  ordi- 

nary   lighting    transformer  —  is    con- 

nected across  the  two  wires  of  the 

telephone    circuit,  and   then   has   its 

middle    point    connected   to  ground, 

the     electrostatic     charge     will     be 

"drained"    off    the  line  without  in- 

terfering  with  the  operation   of  the 

telephone.     The  telephone  line  par- 

alleling the  110,000-  volt  transmission 

of    the  Georgia  Railway  and  Power 

Co.,1  is  provided  with  15-k.w.  stand- 

ard      2200-Volt      distribution      trans-         FIG.  42.—  Telephone  wires  on 
.  ,     .  ,  j      •     \    e         same  pole  as  single  phase  power 

formers  (with   open  secondaries)  for    circuit. 
this  purpose. 

69.  Magnetic  Induction.—  Referring  to  Fig.  42,  the  voltage 
induced  by  the  single-phase  power  circuit  AB  in  the  loop  formed 
by  the  wires  C  and  D  of  the  parallel  telephone  circuit  may  be 
calculated  as  follows  if  we  assume  that  there  are  no  transposi- 
tions and  that  the  current  wave  is  a  pure  sine  curve. 

By  formula  25  (Art.  43)  the  flux  in  the  loop  CD  due  to  the 
conductor  A  carrying  a  current  7  is 


and  the  flux  due  to  the  current  —  /  in  the  conductor  B  is 

1  Refer  to  interesting  article  by  E.  P.  Peck  in  Electrical  World,  Sept.   9, 
1916,  Vol.  68,  p.  515. 


124 


ELECTRIC  POWER  TRANSMISSION 


the  total  flux  being, 


By  making  the  substitutions  and  alterations  as  in  obtaining 
formula  (27)  we  get, 

Volts  induced  per  mile  run  of  the  j   =  Q  Q          ,         /oA\     (53) 
two  parallel  circuits  \aco<i/ 

Since  the  value  of  log  1  is  zero,  there  will  be  no  flux  linkages 
with  the  telephone  circuit  when  the  condition  a<jbc  =  acbd  is 


A    „.-- 


FIG.   43. — Arrangement   of   wire3   which   eliminates   magnetic   induction   from 
telephone  circuit. 

satisfied  (see  Fig.  42).  This  would  be  the  case  in  either  of  the 
arrangements  shown  in  Fig.  43.  It  is  true  that  these  arrange- 
ments refer  to  a  telephone  line  running  parallel  to  a  single-phase 
power  circuit,  and  that  in  any  case  transpositions  would  probably 
be  necessary  in  practice;  at  the  same  time  more  attention  might, 
in  the  writer's  opinion,  be  given  to  the  relative  positions  of  power 
and  telephone  lines,  apart  from  the  question  of  transpositions. 
Practical  methods  of  transposing  both  power  and  telephone 
wires  will  not  be  considered  here,  as  the  number  of  possible  condi- 
tions to  be  remedied  is  almost  unlimited ;  but  once  the  principle 
is  understood,  some  means  of  obtaining  the  required  result  can 
generally  be  found.  Transpositions  on  the  high  tension  power 


ELECTRICAL  PRINCIPLES  AND  CALCULATIONS    125 

lines  should  be  avoided  as  far  as  possible,  and  it  should  not  be 
necessary  to  transpose  the  power  conductors  of  a  high  voltage 
transmission  line  at  more  frequent  intervals  than  every  mile. 

Although  formulas  (52)  and  (53)  were  worked  out,  for  sim- 
plicity, by  taking  the  case  of  a  single-phase  transmission,  the 
inductive  effects  of  any  number  of  power  conductors  maybe 
calculated  in  a  similar  manner,  proper  attention  being  paid 
not  only  to  the  magnitude  but  also  to  the  direction,  or  phase 
relation,  of  the  currents  in  the  several  conductors.  The  general 
problem  of  inductive  effects  between  parallel  wires  is  taken  up 
in  Appendix  I. 

The  fundamental  frequency  (/)  occurs  in  the  formulas,  be- 
cause pure  sine  waves  are  assumed;  but  it  should  be  pointed  out 
that  disturbances  due  to  voltages  and  currents  of  the  funda- 
mental frequency  have  little  effect  on  the  telephone,  the  chief 
trouble  being  due  to  the  fact  that  in  practice  the  pure  sine  waves 
are  not  obtained,  and  the  higher  harmonics,  even  when  of  small 
magnitude,  are  liable  to  produce  noises  in  so  sensitive  an  instru- 
ment as  the  telephone  receiver,  which  may  render  conversation 
impossible. 

It  is  not  proposed  to  discuss  here  the  protective  apparatus 
as  used  on  telephone  circuits  paralleling  power  lines,  partly 
because  the  subject  is  somewhat  beyond  the  scope  of  this  book, 
but  also  because  the  troubles  of  the  telephone  engineer  in  this 
connection  have  not  been  entirely  overcome,  and  he  is  still  work- 
ing on  the  problem.  The  reader  who  desires  to  pursue  the  sub- 
ject further  is  therefore  referred  to  other  sources  of  information 
among  which  may  be  mentioned  the  interesting  Report  of  the 
Joint  Committee  on  Inductive  Interference  to  the  Railroad 
Commission  of  the  State  of  California,  which  will  be  found  on 
page  1441  of  Vol.  XXXIII  (Part  II)  of  the  Transactions  of  the 
American  Inst.  E.  E.,  and  the  paper  by  H.  S.  Warren  entitled 
"Inductive  Effects  of  Alternating  Current  Railroads  on  Com- 
munication Circuits"  in  the  Proceedings  A.  I.  E.  E.  of  Aug., 
1918.1 

70.  Fault  Localizing. — The  location  of  broken  insulators, 
crossed  or  fallen  wires,  or  any  fault  leading  to  unsatisfactory  or 
interrupted  service,  must  be  dealt  with  by  the  operating  staff; 

1  Refer  also  to  the  recent  paper  by  William  W.  Crawford,  "Telephone 
circuits  with  zero  mutual  induction  "  in  the  Proceedings  A.  I.  E.  E.,  p.  377, 
Vol.  XXXVIII,  No.  3,  March,  1919. 


126  ELECTRIC  POWER  TRANSMISSION 

and  any  but  the  briefest  reference  to  these  matters  would  be 
beyond  the  scope  of  this  book. 

The  usual  methods  of  testing  are  explained  in  many  text-books, 
and  in  the  electrical  pocket-books  and  hand-books;  but  the  well- 
known  Varley  and  Murray  loop  tests  are  not  always  satisfactory 
on  high- voltage  transmission  lines.  Then,  again,  the  conditions 
in  regard  to  grounded  or  ungrounded  neutrals,  transformer  con- 
nections, positions  of  section  switches,  telephonic  facilities,  etc., 
are  so  variable  that  it  would  be  a  difficult  matter  to  lay  down 
rules  to  be  followed  in  emergencies,  except  in  connection  with  a 
particular  system;  but  such  rules  should  be  laid  down  by  the 
chief  operating  engineer,  and  rigidly  adhered  to.  By  giving 
careful  attention  to  the  position  of  patrolmen's  houses,  switching 
and  telephone  stations,  as  referred  to  in  Chapter  I,  much  may  be 
done  toward  preventing  long  interruption  of  service  in  the  event  of 
accidents. 


CHAPTER  V 
INSULATION  OF  OVERHEAD  TRANSMISSION  LINES 

71.  Insulator  Materials. — The  material  most  commonly  used 
for  insulators  on  high-tension  overhead  lines  is  porcelain;  but 
glass,  which  is  cheaper  than  porcelain,  may  sometimes  be  used 
to  advantage  on  the  lower-voltage  lines.  Insulators  made  of 
special  moulded  materials  such  as  "  Electrose  "  have  the  advantage 
over  both  glass  and  porcelain  in  that  they  are  lighter  in  weight 
and  less  liable  to  fracture  from  mechanical  shocks  or  high-pressure 
discharges.  This  material  is  used  in  preference  to  porcelain  by 
the  Canadian  Niagara  Power  Co.  on  its  high-voltage  transmission 
lines.  Glass  is  a  material  of  high  resistance  and  dielectric 


Fio.  44. — Single  piece  glass  insulator.    Average  weight  per  piece,  4  j  Ib. 
Made  for  standard  1  in.  and  special  If  in.  pins.     Voltages — Test — Dry  86,200 
Wet  50, 100,  Line  17,000. 

strength  and  makes  excellent  insulators  for  pressures  up  to  about 
25,000  volts.  The  fact  that  it  is  pervious  to  light  tends  to  dis- 
courage spiders  and  cocoon-spinning  insects,  which  ordinarily 
find  their  way  to  the  interior  of  insulators;  but,  on  the  other 
hand,  more  moisture  will  condense  on  the  inside  surfaces  of  glass 
than  on  porcelain  insulators,  thus  attracting  duat  and  dirt. 

Figs.  44  and  45  are  from  drawings  supplied  by  the  Hemingray 
Glass  Co.  of  Muncie,  Ind.  Some  interesting  particulars  relating 
to  the  9-in.  glass  insulator  with  sleeve  (Fig.  45)  have  been  commu- 

127 


128 


ELECTRIC  POWER  TRANSMISSION 


nicated  to  the  writer  by  Mr.  M.  H.  Gerry,  Jr.,  General  Superin- 
tendent of  the  Missouri  River  Electric  and  Power  Co.  of  Helena, 
Montana.  Mr.  Gerry,  who  has  something  like  50,000  of  these 
insulators  on  his  transmission  lines,  is  actually  using  them  for 
working  pressures  up  to  70,000  volts,  where  they  have  given  entire 
satisfaction  for  the  last  13  years.  The  wooden  supporting  pins  are 
treated  with  paraffin  in  a  vacuum,  and  moreover,  the  climate  is 
dry  and  the  air  free  from  dust  or  salt  which  might  cause  trouble 
with  so  high  a  voltage  under  less  favorable  conditions.  Mr. 


FIG.  45. — Two-piece  glass  insulator.     Average  weight,  12f  Ib. 
Voltages— Test— Dry  110,000,  Wet  95,000,  Line  33,000. 

Gerry  sums  up  his  opinion  of  glass  insulators  for  low  and  medium 
voltages  by  stating  that  notwithstanding  their  mechanical  weak- 
ness relatively  to  porcelain,  they  are  entirely  satisfactory  when 
intelligently  used  under  suitable  conditions. 

For  the  suspension  type  of  insulator,  as  used  for  the  higher 
pressures,  porcelain  appears  to  be  the  best  material  available  at 
present,  although  it  is  far  from  being  ideally  suited  to  the  re- 
quirements. An  enormous  amount  of  study  and  research  has 
lately  been  devoted  to  the  development  and  improvement  of 


INSULATION  OF  OVERHEAD  LINES  129 

porcelain  insulators,  and  it  is  not  improbable  that  if  as  much  engi- 
neering skill  had  been  devoted  to  the  improvement  of  glass 
insulators,  these  might  now  be  in  more  general  use,  even  in  the 
form  of  suspension  insulators  for  the  highest  voltages. 

72.  Design  of  Insulators. — The  design  of  an  insulator,  to 
comply  with  any  given  specification,  is  a  matter  which  concerns 
the  manufacturer,  who  has  been  compelled  of  late  years,  owing  to 
the  rapid  increase  of  working  pressures,  to  devote  his  attention 
to  the  principles  underlying  the  correct  and  economical  design  of 
insulators  for  high-pressure  lines,  and  who  is  therefore  something 
of  a  specialist  on  this  particular  subject.  The  transmission-line 
engineer  should  understand  the  principles  underlying  the  correct 
design  of  overhead  insulators;  but  it  is  suggested  that,  however 
great  his  knowledge  of  the  subject,  he  will  be  well  advised  to  leave 
details  of  design  to  the  manufacturer,  and  to  use  standard 
types  when  possible. 

The  design  of  insulators'  for  the  lower  voltages  is  a  compara- 
tively simple  matter,  the  difficulties  becoming  greater  with  the 
increase  of  pressure,  although  the  introduction  of  the  suspen- 
sion type,  which  permits  of  many  units  being  connected  in  series, 
has  considerably  simplified  the  problem. 

It  is  important  to  bear  in  mind  that  every  insulator  is  necessa- 
rily a  more  or  less  complicated  condenser,  and  it  can  generally  be 
thought  of  as  consisting  of  a  number  of  separate  condensers  in 
series,  the  dielectric  being  alternately  air  and  porcelain.  The 
current  passing  from  line  to  ground  is  partly  a  leakage  current  over 
the  surfaces  (the  leakage  through  the  porcelain  being  generally 
negligible),  and  partly  a  capacity  current.  This  capacity 
current  spreads  itself  over  the  high  resistance  surfaces  of  the 
insulator  material  in  a  way  which  will  depend  upon  the  surface 
conductivity  and  on  the  spacing  and  disposition  of  the  various 
parts.  It  is  well  to  keep  the  electrostatic  capacity  as  low  as 
possible,  but  it  is  of  equal  if  not  greater  importance  so  to  dis- 
tribute it  by  a  scientific  arrangement  of  the  component  parts 
that  abnormal  stresses  will  not  occur  locally,  as  these  may  punc- 
ture or  damage  the  insulator  at  one  particular  part,  while  a  more 
carefully  designed  insulator  of  lighter  weight  may  withstand 
a  greater  total  breakdown  pressure,  because  proper  attention 
has  been  given  to  this  important  matter  of  capacity  distribution. 
The  effect  of  rain  on  the  exposed  surfaces  of  an  insulator  is  to 
increase  the  capacity,  and  this  will  generally  lower  the  flash-over 


130  ELECTRIC  POWER  TRANSMISSION 

point;  but  the  increased  surface  conductivity  has  the  effect  of 
equalizing  the  potential  distribution;  and  in  the  case  of  a  large 
number  of  condensers  in  series,  such  as  occurs  especially  with  the 
suspension  type  of  insulator,  it  has  actually  been  observed  that 
this  equalizing  of  the  potential  distribution  may  cause  the  flash- 
over  pressure  of  the  wet  insulator  to  be  no  lower  than  the  flash- 
over  pressure  of  the  same  insulator  when  dry. 

If  the  distribution  of  dielectric  flux  could  be  easily  determined 
in  the  case  of  the  rather  complicated  shapes  and  varying  thick- 
nesses of  dielectric  which  occur  in  high  tension  line  insulators,  it 
would  be  an  easy  matter  to  predict  the  performance  of  new  types 
and  sizes  under  specified  conditions;  but  although  the  dielectric 
circuit  can  conveniently  be  treated  in  a  manner  analogous  to  the 
engineer's  treatment  of  the  magnetic  circuit,  there  is  always 
difficulty  (except  in  the  simplest  cases)  in  predetermining  the 
amount  and  direction  of  the  lines  of  flux  —  or  stress. 

The  fundamental  law  of  the  dielectric  circuit  is 

*  =  EC  (54) 

where  V  is  the  total  dielectric  flux  (in  coulombs)  in  a  space  of 
which  the  permittance  or  capacity  is  C  farads,  when  the  potential 
difference  producing  this  flux  is  E  volts. 

If  we  consider  a  small  element  of  flux,  i.e.,  a  "tube"  of  induc- 
tion, of  length  I  cm.  and  cross-section  A  sq.  cm.  over  which 
the  flux  SF  is  evenly  distributed,  the  flux  density,  in  coulombs 
per  square  centimeter,  is 

D  =  */A  (55) 


If  the  difference  of  potential  between  the  two  ends  of  the  path 
I  cm.  long  is  E  volts,  the  capacity  (or  permittance)  is  propor- 
tional to  A/I  —  exactly  as  in  the  analogous  case  of  the  magnetic 
circuit  of  which  the  permeance  is  directly  proportional  to  A  and 
inversely  proportional  to  I. 

With  the  introduction  of  the  proper  constants,  we  have, 

I-  A 

C  =  K™  (56) 

where  K  has  the  numerical  value  8.84  X  10~u  and  k  is  the 
specific  inductive  capacity  or  dielectric  constant  of  the  material. 
Approximate  values  for  k  are  given  in  the  accompanying 
table. 


INSULATION  OF  OVERHEAD  LINES  131 

DIELECTRIC  CONSTANTS  AND  DISRUPTIVE  VOLTAGES 


Material 

Dielectric  constant  k 

Dielectric  strength, 
k.v.  per  cm. 

Air 

1 

22 

Porcelain 

4  4  to  6 

100 

Glass 

5  to  10 

90 

Transformer  oil 

2  2to  2  5 

Paraffin  

1.9to2.3 

100 

The  figures  in  the  last  column  of  the  Table  are  approximate 
only:  they  indicate  the  virtual  orr.m.s.  value  of  the  (sinusoidal) 
alternating  voltage  which  may  be  expected  to  break  down  a  slab 
of  the  material  1  cm.  thick  placed  between  two  large  flat  metal 
electrodes.  What  is  usually  understood  by  the  expression  "dis- 
ruptive gradient"  is  obtained  by  multiplying  the  values  in  the 
Table  by  \/2. 

A  more  useful  expression  for  the  flux  density,  D,  can  now  be 
obtained  by  putting  the  value  of  C  from  (56)  in  formula  (54) 
and  substituting  the  value  of  ^  so  obtained  in  formula  (55). 
Thus 


but  E/l  is  the  potential  gradient,  or  volts  per  centimeter  —  usually 
denoted  by  the  letter  G  —  whence 


D  =  Kk  X  G 


(57) 


This  clearly  indicates  that,  in  a  given  material,  the  potential 
gradient  (G)  is  directly  proportional  to  the  dielectric  flux  density 
(D),  and  if  we  can  estimate  the  one,  we  can  calculate  the  other. 
The  chief  problem  in  insulator  design  is  so  to  proportion  the  parts 
that  the  flux  density  shall  nowhere  be  so  great  as  to  cause  break- 
down of  the  air  or  puncture  of  the  solid  material  of  the  insulator. 
An  example  illustrating  the  application  of  the  formulas  will  be 
given  in  connection  with  the  design  of  wall  bushings,  as  this  is  a 
somewhat  simpler  problem  than  the  design  of  high  voltage  line 
insulators. 

73.  Pin-type  Insulators. — As  previously  mentioned,  the  sheds 
or  petticoats  with  intervening  air  spaces  which  separate  the  wire 
at  line  potential  from  the  pole  or  cross-arm  which  is  usually  at 
ground  potential,  may  be  thought  of  as  a  number  of  condensers 


132  ELECTRIC  POWER  TRANSMISSION 

in  series.  If  Ic  is  the  charging  current,  and  E  is  the  potential 
difference  causing  this  flow  of  current  through  a  condenser  of 
capacity  C,  we  have  the  relation  Ic  =  2irfEC;  and,  when  the 

frequency  is  constant,  E  oc  -^  -  Thus,  when  a  number  of  con- 
densers are  connected  in  series,  the  current  Ic  is  the  same  through 
all  the  condensers,  and  the  potential  difference  across  any  one 
condenser  is  inversely  proportional  to  its  capacity;  whence  the  im- 
portance of  care  in  design  to  avoid  too  great  a  stress  across  a 
given  thickness  of  porcelain.  With  the  pin  type  of  insulator, 
a  number  of  sheds  or  petticoats  hanging  close  to  the  pin,  with 
small  air  spaces  between  them,  will  not  be  effective,  because, 
although  the  leakage  path  may  be  long,  the  capacity  is  high. 
The  remedy  consists  in  spreading  the  petticoats  away  from  the 
pin,  the  outer  shed  in  some  designs  being  almost  horizontal. 
This  outer  shed  has,  in  some  cases,  been  replaced  by  a  metal 
shield.  Apart  from  the  advantage  of  lightness,  which  permits 
of  a  thin  metal  shed  being  made  of  larger  diameter  than  would 
be  permissible  if  the  material  were  porcelain,  the  charging  current 
will  spread  itself  more  uniformly  over  the  surface  of  the  outer 
shed,  and  so  prevent  the  concentration  of  potential  at  the  point 
where  the  conductor  is  tied  to  the  insulator.  An  insulator  of 
this  type  may  flash  over  at  a  somewhat  lower  pressure  than  if 
the  upper  hood  were  of  porcelain,  but  under  wet  conditions,  the 
flash-over  pressure  may  be  higher. 

Pin-type  insulators  are  available  for  pressures  up  to  70,000 
volts,  but  the  suspension  type,  made  up  of  two  of  more  units, 
will  generally  be  more  satisfactory  and  economical  for  pressures 
above  50,000  volts.  The  pin  type  of  insulator  becomes  too  heavy 
and  costly  when  designed  for  the  higher  voltages,  and,  owing  to 
the  great  length  of  the  supporting  pin,  the  bending  moment  near 
the  point  of  attachment  to  the  cross-arm  tends  to  become 
excessive. 

Wooden  supporting  pins  are  not  recommended  for  high-tension 
work;  they  are  rarely  used  on  lines  working  at  pressures  above 
33,000  volts.  Metal  pins  are  generally  preferable. 

Fig.  46  shows  the  sparking  distances  as  usually  measured. 
The  dotted  line  on  the  left  side  of  the  drawing  shows  the  length  of 
path  on  a  dry  flash-over.  When  the  exposed  surfaces  of  the  in- 
sulator are  wet  with  rain,  these  surfaces  are  looked  upon  as  con- 


INSULATION  OF  OVERHEAD  LINES 


133 


ductors,  and  the  flash-over  distance  is  the  sum  of  the  separate 
distances  A,  B,  and  C;  the  lines  A  and  B  being  usually  drawn  at 
an  angle  of  45  degrees  to  the  vertical.  In  this  particular  case, 
the  line  C  is  also  drawn  at  45  degrees  because  the  metal  pin  is 
surrounded  by  a  porcelain  base.  The  length  of  the  pin  should  be 
such  that  the  distance  E  to  cross-arm  is  slightly  greater  than  the 
sparking  distance  C  from  edge  of  inner  petticoat  to  pin. 

The  flash-over  distances  as  measured  on  actual  insulators  of  the 
pin  type  are  approximately  as  given  below.1 


FIG.  46. — Sparking  distances  on  pin 
type  insulator. 


FIG.  47. — Pin  type  insulator. 


Voltage,  r.m.s.  value 


Flash-over  distance,  in. 


40,000  

3 

60  000 

4M 

80,000 

6% 

100,000                                      .      .      .. 

8M 

120,000  

11 

140,000  

14 

160000 

18K 

Figs.  47,  48,  and  49  are  illustrations  of  typical  pin  type  insula- 
tors as  made  by  R.  Thomas  &  Sons  of  East  Liverpool,  Ohio. 

1  Average  figures  taken  from  curves  given  by  Mr.  J.  Lustgarten  in  the 
Jour,  of  the  Inst.  E.  E.,  vol.  49  (1912),  p.  235. 


134 


ELECTRIC  POWER  TRANSMISSION 


The  sections  shown  in  Figs.  48  and  49  illustrate  the  trend  in 
design  of  modern  pin-type  insulators  toward  smaller  height  in 
proportion  to  the  head  diameter.  The  same  may  be  said  of 
Figs.  50  and  51  which  illustrate  new  standard  types  of  "Victor" 
insulator  manufactured  by  Locke  Insulator  Manufacturing  Co. 
of  Victor,  N.  Y.  A  tendency  to  increase  the  thickness  of  porce- 
lain is  noticeable  in  all  recent  designs  of  line  insulators.  The  lead- 


Fio.  48. 


FIG.  49. 


Pin  type  insulator. 


ing  particulars  of  these  pin-type  insulators,  as  furnished  by  the 
makers,  are  as  follows: 


Fig.  47 

Fig.  48 

Fig.  49 

Fig.  50 

Fig.  51 

Line  pressure,  volts  

25,000 

25,000 

45,000 

45,000 

50,000 

Dry  flash-over,  volts 

80,000 

89000 

147  000 

150  000 

175  000 

Wet  flash-over,  volts 

50,000 

54,000 

102,000 

90,000 

120,000 

Leakage  distance,  inches  .... 
Arcing  distance  (wet),  inches 
Net  weight,  each,  pounds.  .  . 

12.5 
4.5 
4.5 

12.0 
5.0 
6.5 

28.00 
7.25 
20.00 

20.0 
13.5 

23.5 

24.5 

74.  Suspension-type  Insulators. — With  the  suspension  type 
of  insulator,  the  conductor  is  hung  below  the  point  of  support 
(which  is  usually  grounded)  at  the  end  of  a  string  of  insulator 
units  connected  to  one  another  by  metal  links.  The  potential 
difference  which  will  cause  a  flash-over  or  breakdown  on  such 
a  series  of  insulators  will  not  be  in  direct  proportion  to  the  number 
of  insulators  in  the  string.  This  is  due  to  the  unequal  distribu- 
tion of  the  potential  differences,  which  is  again  a  question  of 
relative  capacities.  The  design  of  the  individual  units  may  ap- 


INSULATION  OF  OVERHEAD  LINES 


135 


pear  to  be  good,  and  yet  a  string  of  such  insulators,  if  these  are 
not  specially  designed  to  fulfil  certain  requirements,  may  give 
surprisingly  unsatisfactory  results.  A  factor  of  importance  is  the 

mutual  capacity        ..,-*_. 
ratl°  capacity  to  ground  whlch  determmes  the  Potential  distil- 

bution;  and  this  ratio  will  depend  not  only  on  the  shape  and  size 
of  the  porcelain  units,  but  also  on  the  metal  caps  or  means  of 
attachment,  and  the  spacing  between  units. 

The  distribution  of  the  potential  drop  on  a  string  of  suspension 
insulators  is  easily  calculated  if  certain  assumptions  are  made 
with  a  view  to  simplifying  the  work.  If  symbols  are  used  to  de- 
note the  various  voltages  and  currents,  the  formulas  become 
somewhat  complicated  in  appearance  although  actually  simple  to 


FIG.  50. — Pin  type  insulator. 


Fio.  51. — Pin  type  insulator. 


apply.  A  very  clear  explanation  of  the  voltage  distribution  on 
the  string  of  suspension  units  has  been  given  by  Mr.  F.  W.  Peek 
Jr. x  The  same  treatment  of  the  problem  although  not  connected 
with  Mr.  Peek's  name,  will  be  found  in  the  Handbook  on  Over- 
head Line  Construction  issued  by  the  National  Electric  Light 
Association.  A  numerical  example  will  best  illustrate  the  manner 
in  which  the  relative  values  of  the  several  capacities  affect  the 
distribution  of  potential  between  the  high-tension  conductor 
and  the  point  of  attachment  to  the  (grounded)  cross-arm. 

The  metal-work  (cap  and  link  or  bolt)  on  each  side  of  the  porce- 
lain disc  constitutes  one  terminal  of  a  condenser  which  has  a 
capacity  C  farads  relatively  to  ground  or  grounded  tower, 

1  "Electrical  Characteristics  of  the  Suspension  Insulator,"  Trans.,  A.  I. 
E.  E.,  p.  907,  vol.  xxxi,  May,  1912. 


136 


ELECTRIC  POWER  TRANSMISSION 


and  a  capacity  C"  =  mC  relatively  to  the  metal-work  (cap  and 
link  or  bolt)  dn  either  side  of  the  porcelain  disc.  This  is  shown 
diagrammatically  in  Fig.  52  where  a,  6,  c,  and  d,  are  four  imagi- 
nary condensers,  each  of  capacity  C  farads,  between  the  ground 
and  the  insulating  metal  caps,  etc.,  while  the  condensers  of  ca- 
pacity mC  farads,  formed  by  each  individual  unit,  are  numbered 
(1),  (2),  (3),  and  (4).  We  shall  neglect  the  effects  of  surface  leak- 
age and  corona.  The  former  would  tend  to  equalize  the  potential 


rr 


FIG.   52. — Diagrammatic  representation   of  4-unit  suspension   type  insulator. 

drop  across  the  insulator  units,  while  the  latter — in  addition  to 
causing  actual  leakage  losses  through  the  air — might  alter  the 
capacity  of  the  units  subjected  to  the  higher  pressures.  No 
error  of  appreciable  magnitude  is  likely  to  be  made  by  neglect- 
ing these  items. 

Assuming  the  voltage  across  the  unit  nearest  to  the  grounded 

mutual  capacity 


cross-arm  to  bo  Ei  =  10,000  volts,  and  the  ratio 
to  be  m  =  10,  we  have, 


capacity  to  ground 


INSULATION  OF  OVERHEAD  LINES  137 

Ei  =  10,000  volts 

/!  =-2*fEi(mC)  =  co  X  10,000  X  10  X  C  • 

Ia  =  2irfEiC  =  co  X  10,000  X  C 
whence 

72  =  (/i  +  /«)  =  cotfiCm  +  1)C  =  co  X  10,000  X  11  X  C 
and 

#2  =  #1  (^)  =  #1  X  ~  =  J  1,000  volts 

Ib  =  co(#i  +  #2)C 
whence 

/3  =  (/a  +  /*)  =  «[#i(m  +  2)  +  #2]C  =  co  X  131,000  X  C 
and 

E3  =  Ei(^=  10,000X^=13,100  volts 

7C  =  u(Ei  +  #2  +  #3)C 
whence 

/4  =  (/s  +  /c)  =  coLE^m  +  3)  +  2E2  +  ^3]C  =  co  X  165,100  X  C 
and 

E,  =  B,  @  =  10,000  X  1^0  .  ,6iHO  volt, 

The  total  potential  difference  across  the  string  of  four  insulators 
is  therefore  not  4Ei  =  40,000,  but  #  =  #1  +  #2  +  #3  +  #4  = 
50,610  volts. 

The  stress  across  the  insulator  (No.  4)  nearest  to  the  high- 
tension  conductor  is  E*  =  16,510.  Assume  that  it  will  flash  over 
with  twice  this  pressure,  or  2E*  =  33,020  volts;  then  the  pressure 
which  will  start  a  flash-over  across  the  string  of  four  insulators 
is  2E  or  101,220  volts,  which  is  less  than  four  times  2#4  or  132,080 
volts.  The  term  "  string  efficiency  "  has  been  appliedby  Mr.  F.  W. 

.    arc-over  voltage  of  string  of  n  insulators 
Peek,  Jr.'  to  the  ratio  n  x  arc.over  voltage  of  single  insulator 

El 

or  —pr  which,  in  this  example,  is 
nHin 

E  50,610     =n?fi. 


4#4      4X16,510 

In  this  connection  it  is  assumed  that  all  the  insulator  units  are 
of  the  same  size  and  shape. 

The  fact  that  the  ratio  (m)  of  the  mutual  capacity  to  the  ca- 
pacity between  the  suspension  link  and  ground  is  an  important 

1  Trans.  A.  I.  E.  E.,  vol.  xxxi,  p.  907,  May,  1912. 


138 


ELECTRIC  POWER  TRANSMISSION 


factor  in  determining  the  distribution  of  potential  over  the  in- 
sulator string  suggests  the  importance  not  only  of  the  shape  and 
surface  area  of  the  metal  fixtures  on  the  indiyidual  insulators, 
but  also  the  length  of  the  connecting  link,  or  spacing  between 
insulators. 

In  actual  tests  made  by  Mr.  Peek,  the  dry  arc-over  required 
1%  times  the  pressure  of  the  wet  arc-over  on  a  single  unit; 
but,  on  a  string  of  nine  such  units,  the  wet  and  dry  flash-overs 
occurred  at  the  same  voltage.  The  "string  efficiencies"  of  these 
insulators  on  the  dry  test  were  as  follows : 


No.  of  units  in  series 

Flash-over 
voltage 

Efficiency, 
per  cent. 

1  
2  

74,000 
137,000 

100.0 
93  0 

3  

186,000 

84.0 

4 

238,000 

80  5 

5 

281,000 

76  0 

6 

318,000 

72  0 

Another  series  of  tests,  quoted  by  Mr.  W.  T.  Taylor,1  gives 
efficiencies  as  below : 


No.  of  units  in  series 

Flash-over 
dry  test 

String 
efficiency  dry 

Flash-over 
wet  test 

String 
efficiency  wet 

1 

90000 

100   0 

56000 

100  0 

2 

160000 

89  0 

90  000 

80  0 

3. 

220,000 

81  5 

130,000 

77  5 

4  

274,000 

76.0 

175,000 

78.0 

5  

310,000 

69.0 

220,000 

78.5 

6 

340  000 

63  0 

265000 

79  0 

Arc-over  voltages  for  strings  of  insulators  are  now  usually 
given  in  manufacturers'  catalogues. 

When  the  mutual  capacity  is  small  relatively  to  the  ground 
capacity,  there  soon  comes  a  point  beyond  which  it  is  useless  to 
put  more  insulators  in  the  string.  Short  strings,  of  well-designed 
and  properly  spaced  units,  will  often  be  more  effective  and  less 
costly  than  a  longer  series  of  insulators  of  which  the  single  units 
may  have  excellent  insulating  properties,  but  may  not  have 

1  Journal  Inst.  E.  E.,  vol.  46,  p.  510  (1911). 


INSULATION  OF  OVERHEAD  LINES  139 

been  specially  designed  for  the  particular  requirements.  It  is 
sometimes  possible  to  increase  the  arc-over  voltage  of  a  string  of 
insulators  by  reducing  the  distance  between  consecutive  units; 
a  fact  that  it  is  difficult  to  understand  unless  the  importance 
of  the  proper  capacity  distribution  has  been  realized.  A  well- 
arranged  string  of  properly  designed  units  not  necessarily  similar 
in  shape  and  size,  is  less  liable  to  damage  by  lightning  and  kindred 
phenomena  than  a  series  of  insulators  of  low  "string  efficiency." 

It  is  well  to  bear  in  mind  that  an  insulator  may,  and  does, 
behave  differently  when  subjected  to  high-frequency  charges  as 
induced  by  lightning  disturbances,  than  when  the  difference  of 
potential  is  applied  at  the  normal  frequency  of  the  transmission 
line.  The  distribution  of  potential  among  a  number  of  condensers 
in  series  should  be  independent  of  frequency,  but  a  high-tension 
insulator  consists  of  many  capacities  in  series  with  resistances 
such  as  the  leakage  paths  through  the  body  of  the  material  and 
over .  the  surfaces,  and  the  distribution  of  the  total  potential 
drop  across  a  condenser  and  resistance  in  series,  even  if  the  latter 
may  be  considered  non-inductive,  is  not  the  same  at  high 
as  at  low  frequencies.  This  is  probably  one  of  the  chief  reasons 
why  insulators  that  will  flash  over  rather  than  puncture  on  tests 
conducted  at  the  ordinary  line  frequency,  will  sometimes  fail 
through  puncture  during  atmospheric  electrical  disturbances.1 

Apart  from  the  great  advantages  from  the  point  of  view  of 
insulation,  which  are  obtained  by  suspending  the  conductor  from 
a  string  of  insulators  connected  in  series,  this  arrangement,  as 
now  generally  adopted  for  pressures  above  50,000  volts,  has  the 
further  advantage  that  the  conductor  is  less  liable  to  be  affected 
by  lightning  disturbances,  since,  at  every  point  of  support,  the 
wire  is  hung  below  the  attachment  to  the  supporting  tower 
which,  in  almost  every  instance  is  a  well-grounded  steel  structure. 
Another  advantage  is  the  comparative  flexibility  of  the  attach- 
ment, which  very  considerably  diminishes  the  possibility  of 
crystallization  of  the  conductor  material,  such  as  is  liable  to 
occur  when  the  wire  is  rigidly  attached  to  the  pin  type  of  in- 
sulator; this  effect  being  more  noticeable  with  aluminum  than 
with  copper. 

In  designing  cross-arms  for  the  attachment  of  the  suspension 
type  of  insulator,  it  is  not  necessary  to  pay  much  attention  to 

1  See  "High  Frequency  Tests  on  Line  Insulators,"  by  L.  E.  Imlay  and 
Percy  H.  Thomas  in  the  Trans.  A.  I.  E.  E.,  vol.  xxxi,  p.  2121. 


140 


ELECTRIC  POWER  TRANSMISSION 


torsional  stresses,  which,  however,  must  be  carefully  considered 
when  pin- type  insulators  are  used  on  heavy  long-span  lines; 
but,  on  the  other  hand,  taller  towers  are  necessary  with  the  sus- 
pension type  of  insulator.  Another  possible  disadvantage  is  the 
wider  spacing  between  wires  generally  adopted  when  this  type 
of  insulator  is  used;  and  it  is  certain  that  rather  more  skill  and 
experience  are  necessary  in  the  stringing  of  the  conductors  than 
when  pin-type  insulators  are  used.  It  is  usual  to  sling  the  wire 


.Sanded 
Cork  Di 


Fia.  53. — Suspension  type  insulator. 


One  unit 

Number  of  units  in  string 

Ultimate 

mechan- 

Leakage 

ical 

distance 

distance 

1 

2 

3 

4 

5 

6 

7 

strength 

10" 

4H' 

Length  of  string  5%" 
Dry  flash-over,  K.  V.  62 

124  * 

16H" 

21M* 
240 

292X 

340  * 

37*i" 
383 

9000  Ib. 

Wet  flash-over,  K.  V.  44 

82 

120 

158 

196 

235 

274 

Net  weight,  Ib  
Gross  weight,  Ib  

io« 

21 

s* 

42 
56 

52^ 
69 

63 

83 

£* 

in  the  first  instance  from  snatchblocks  attached  to  the  cross-arms 
of  every  tower  over  a  distance  of  about  a  mile,  or  between 
anchoring  or  corner  towers,  and  then  draw  up  tight;  the  con- 
ductor being  subsequently  transferred  at  every  point  of  attach- 
ment from  the  snatchblock  to  the  permanent  suspension  clamp 
at  the  end  of  the  lowest  insulator  in  the  series. 

In  the  interlink  type  of  suspension  insulator,  the  porcelain 


INSULATION  OF  OVERHEAD  LINES 


141 


between  the  metal  links  is  in  compression,  and  in  the  event  of  the 
shattering  of  the  porcelain  parts,  the  conductor  remains  sus- 
pended; but  although,  from  the  mechanical  point  of  view,  this 
type  is  to  be  recommended,  it  is  more  liable  to  puncture  owing 
to  excessive  electric  stress  in  the  thickness  or  porcelain  between 
the  metal  links,  than  the  more  usual  type  of  insulator  with  cap 
and  bolt  cemented  to  the  porcelain  parts  as  shown  in  the  accom- 
panying illustrations. 


FIG.  54. — Suspension  type  insulator. 


One  unit 

Number  of  units  in  string 

Ultimate 

mechan- 

Leakage 
distance 

Wet 

arcing 
distance 

1 

2 

3 

4 

* 

6 

7 

ical 

strength 

ion' 

5>i" 

Length  of  string  
Dry  flash-over,  K.  V. 
Wet  flash-over,  K.  V. 

80^' 
49 

20H" 
160 
103 

30  H" 
239 
157 

41" 
311 
211 

a$i*' 

265 

61>£" 
440 
319 

26,000:ib. 

Net  weight,  Ib  

31 

62 

93 

124 

155 

186 

78 

g 

Figs.  53  and  54  are  sectional  views  of  suspension  insulators 
manufactured  by  the  R.  Thomas  and  Sons  Co.  of  East  Liverpool, 
Ohio;  and  Fig.  55  is  an  insulator  made  by  the  Ohio  Brass  Co. 
of  Mansfield,  Ohio,  as  originally  selected  for  the  140,000-volt 
transmission  lines  of  the  Au  Sable  Electric  Co.  (Au  Sable  to 
Battle  Creek,  Mich.),  a  string  of  10  units  being  used  on  this 
voltage. 


142 


ELECTRIC  POWER  TRANSMISSION 


A  great  deal  of  attention  has  recently  been  devoted  to  the  im- 
provement of  the  suspension  type  of  insulator.  The  parts  of  the 
early  designs  were  cemented  solidly  together,  with  the  result  that 
the  differential  expansion  of  porcelain  and  metal  caused  cracking 
and  consequent  destruction  of  the  procelain  after  being  in  use 
for  only  a  few  years.  That  experts  in  insulator  design  realize 
the  effects  of  expansion  and  other  causes  leading  to  rapid  deterio- 
ration of  porcelain  insulators  is  evidenced  by  the  experimental 
work  that  has  been  carried  on,  and  the  number  of  papers  pub- 
lished, relating  to  this  subject,  during  the  last  two  or  three  years. 


c > 

FIG.  55. — Suspension  type  insulator. 


Fig.  56,  reproduced  from  a  drawing  kindly  supplied  by  the 
Locke  Insulator  Manufacturing  Co.,  shows  how  allowance  has 
been  made  for  expansion.  An  insulator  unit  as  shown  in  Fig. 
56  may  be  used  in  both  vertical  and  horizontal  positions.  When 
heavy  strains  have  to  be  resisted,  it  is  often  preferable  to  use  two 
or  three  strings  of  insulators  in  multiple.  A  good  design  of 
suspension  unit  can  be  made  to  have  a  breaking  strength  of  about 
6000  lb.,  but  for  greater  mechanical  strength  some  of  the  best 
features  of  design  from  the  electrical  viewpoint  may  have  to  be 
sacrificed. 

An  interesting  discussion  of  the  modern  line  insulator  will  be 


INSULATION  OF  OVERHEAD  LINES 


143 


found  in  the  Paper  by  Mr.  G.  V.  Twiss  in  Vol.  55,  No.  267  (June, 
1917)  of  the  Journal  of  the  Institution  of  Electrical  Engineers. 
It  is  generally  true  that  American  engineers  are  the  acknowledged 
experts  in  high-tension  transmission  work,  and  so  far  as  overhead 
lines  in  England  are  concerned,  these  are  so  few  and  of  such  low 
voltage  that  they  need  not  be  considered.  But  British  engineers 
have  carried  out  important  developments  in  India,  Africa,  and 
other  Colonies  abroad,  and  such  work,  although  not  invariably 
based  on  American  practice,  is  always  carefully  planned  and  exe- 
cuted. The  differences  between  British  and  American  methods 
are  worthy  of  some  study,  if  only  to  put  us  on  our  guard  against 
the  tyranny  of  "established  practice." 


FIG.  56. — Suspension  type  insulator. 
Line  voltage  =  16,000.      Leakage  distance  =  12  in.     Striking  distance  =  3  J  in. 

75.  Wall  and  Roof  Outlets. — When  overhead  high-tension  con- 
ductors have  to  be  brought  into  buildings,  the  design  of  insulating 
bushings  should  receive  careful  consideration.  The  fact  that 
every  bushing  acts  as  a  condenser  or  series  of  condensers  between 
wire  and  ground,  must  steadily  be  recognized  just  as  in  the  case 
of  the  pole-line  insulators,  except  that  the  effect  is  even  more 
marked  in  connection  with  bushings  entirely  surrounding  the 
conductor  than  with  insulators  that  support  the  wire  at  one  end 
only.  When  the  climate  and  weather  conditions  are  favorable, 
it  is  well  to  avoid  bushings  entirely.  In  such  cases,  the  wires 
cannot  be  brought  down  through  the  roof  of  the  building,  but 


144 


ELECTRIC  POWER  TRANSMISSION 


they  must  enter  at  the  side;  a  suitable  protecting  hood  or  roof 
being  placed  above  the  wires  on  the  outside  of  the  building. 

The  smallest  dimension  of  the  opening  in  brick,   stone,  or 
concrete  wall  should  preferably  not  be  less  than  as  given  below: 


For  line  pressure,  volts 


22,000 
33;000 
44,000 
66,000 
88,000 
110,000 


On  each  side  of  the  wall  opening,  the  conductor  is  carried  by 
line  insulators,  of  the  pin  or  suspension  type,  as  the  voltage  may 


Fio.  57. — Porcelain  entering  bushing  for  70,000  volts. 

require,  these  being  so  arranged  as  to  maintain  the  conductor  in 
the  center  of  the  opening,  with  a  slight  downward  incline  toward 
the  outside  of  the  building  to  prevent  rain-drops  being  carried  to 
the  inside. 

When  bushings  are  used,  these  must  necessarily  be  thicker  at  or 
near  the  center,  where  the  ground  potential  is  brought  up  close 
to  the  conductor,  than  at  either  end.  Fig.  57  shows  a  porcelain 
bushing  built  up  of  four  parts,  as  supplied  by  the  R.  Thomas  & 


INSULATION  OF  OVERHEAD  LINES 


145 


Sons  Co.,  suitable  for  a  working  pressure  of  70,000  volts.  This 
bushing  weighs  about  100  Ib.  and  will  flash  over  with  170,000 
volts  (dry)  and  138,000  volts  (wet).  Fig.  58  shows  a  floor  bushing 
suitable  for  a  working  pressure  of  44,000  volts.  Its  dry  flash-over 
voltage  is  112,000,  and  wet  flash-over  75,000. 

One  method  of  bringing  the  high  tension  conductor  through 
the  wall  of  a  building  is  illustrated  by  Fig.  59,  which  shows  the 
overhead  wire  anchored  to  the  wall  by  a  string  of  suspension 
insulators,  and  passing  through  a  standard  porcelain  wall  bush- 
ing as  manufactured  by  the  Ohio  Brass  Co. 


Fia.  58. — Floor  bushing. 

In  the  case  of  a  roof  bushing,  permitting  of  the  conductor 
passing  vertically  downward  to  the  inside  of  the  building,  a  hollow 
elongated  barrel-shaped  insulator  (that  is,  of  greater  diameter  at 
center,  where  it  is  supported  on  the  outside,  than  at  the  two  ends) 
filled  with  an  insulating  substance  of  higher  specific  induct- 
ive capacity  and  greater  dielectric  strength  than  air,  makes  a 
satisfactory  arrangement  provided  there  is  no  danger  of  the  oil  or 
other  insulating  filling  leaking  out  of  the  containing  shell.  A  high- 
grade  insulating  oil  will  frequently  give  good  results,  but  it  is  liable 
to  leak  out  at  the  joints,  and,  moreover,  the  use  of  oil  calls  for  a 


146 


ELECTRIC  POWER  TRANSMISSION 


larger  and  heavier  bushing  than  if  the  filling  has  a  specific  ca- 
pacity more  nearly  equal  to  that  of  the  porcelain  shell  (porcelain 
being  the  material  most  generally  used).  It  will  be  understood 


Hood  should  be  used  where. 
Icicles  can  fall  on  Insulators., 
from  Eaves 


FIG.  59. 


that  if  the  substance  filling  the  space  between  metal  conductor 
and  hollow  bushing  has  the  same  specific  capacity  as  the  material 
of  the  bushing,  there  will  be  no  change  in  the  potential 
gradient  at  the  inner  surface  of  the  bushing.  The  thickness  of 

the  porcelain  shell  and  the 
distance  between  inner  surface 
of  shell  and  conductor  surface 
should  be  proportioned  in  ac- 
cordance with  the  insulating 
material  intended  for  use  as  a 
filling. 

76.  Design  of  Insulating 
Bushings. — Without  attempt- 
ing to  go  into  the  details  of  de- 
sign, the  application  of  the 
principles  and  formulas  of  Ar- 
ticle 72  may  be  illustrated  by 
considering  the  stresses  in  the 
insulation  which  separates  a 

cylindrical  rod  at  high  potential  from  a  concentric  cylindrical 
tube  at  ground  potential.  Fig.  60  is  a  section  through  a  con- 
ductor of  radius  r  separated  by  insulating  material  of  specific 


Fio.  60. — Section  through  insulating 
bushing. 


INSULATION  OF  OVERHEAD  LINES  147 

inductive     capacity    k    from    a    concentric    metal    cylinder    of 
radius  R. 

The   equipotential   surfaces   will  be   cylinders,   and   the   flux 

density  over  the  surface  of  any  cylinder  of  radius  x  and  of  unit 

\fr 
length,  say  1  cm.,  will  be  D  =  ~  -- 

By  formula  (57)  the  potential  gradient  is, 


In  order  to  express  this  relation  in  terms  of  the  total  voltage 
E,  it  is  necessary  to  substitute  for  the  symbol  ^  its  equivalent 
E  X  C,  and  calculate  the  capacity  C  of  the  condenser  formed  by 
the  rod  and  the  concentric  tube.  Considering  a  number  of  con- 
centric shells  in  series,  the  elastance1  may  be  written  as  follows: 


i-  r  ^ 

C      J    2irxKk 


dx  1     ,       R  /rr.x 

e-  (59) 


Substituting  in  (58),  we  have 
E 


volts  per  centimeter  .     . 


the  maximum  value  of  which  is  at  the  surface  of  the  inner  con- 
ductor, where 

--  (61) 


This  formula  is  of  some  value  in  determining  the  thickness  of 
insulation  necessary  to  avoid  overstressing  the  dielectric;  but  it 
is  not  strictly  applicable  to  wall  bushings  in  which  the  outer 
metal  surface  is  short  as  compared  with  the  diameter  of  the 
opening.  The  advantage  of  having  a  fairly  large  value  for  r  is 
indicated  by  formula  (61),  and  a  good  arrangement  is  to  use  a 
hollow  tube  for  the  high-tension  terminal. 

Solid  porcelain  bushings  with  either  smooth  or  corrugated 
surfaces  may  be  used  for  any  pressure  up  to  about  40,000  volts. 
In  designing  plain  porcelain  bushings  it  is  important  to  see  that 
the  potential  gradient  in  the  air  space  between  the  metal  rod  and 

1  The  reciprocal  of  the  permittance  or  capacity. 


148  ELECTRIC  POWER  TRANSMISSION 

the  insulator  is  not  liable  to  cause  brush  discharge,  as  this  would 
lead  to  chemical  action,  and  a  green  deposit  of  copper  nitrate  upon 
the  rod. 

Fig.  61  shows  a  bushing  that  is  liable  to  give  trouble  owing 
to  brush  discharge  in  the  air  space  between  the  high-tension  rod 
and  the  inner  surface  of  the  porcelain  bushing.  The  reason  for 
this  will  be  best  understood  by  working  out  a  numerical  example. 
Let  us  assume  the  following  numerical  values: 

Radius  of  inner  conductor,  r  =  0.25  in. 

Inside  radius  of  porcelain  bushing,  r'  =  0.375  in. 

Outside  radius  of  porcelain  bushing  (in  contact  with  grounded 
metal  cylinder)  R  =  1.5  in. 

Dielectric  constants:  for  air  k  =  1;  for  porcelain  k  =  4.5. 


FIG.  61. — Type  of  bushing  liable  to  cause  brush  discharge  on  surface   of  wire. 

We  have  here  the  case  of  two  capacities  in  series,  the  first  between 
the  high-tension  rod  and  the  inner  surface  of  the  porcelain,  with 
air  as  the  dielectric,  and  the  second  between  the  inner  and  outer 
cylindrical  surfaces  of  the  porcelain.  The  dielectric  flux  (and 
the  charging  current)  being  the  same  in  both,  and  since  ^  = 
E  X  C,  it  follows  that  the  potential  difference  across  each  con- 
denser will  be  inversely  proportional  to  the  capacity,  or  directly 
proportional  to  the  elastance  as  given  by  formula  (59). 

Let  Ea  be  the  voltage  across  the  air  gap,  and  Ep  the  voltage 
across  the  porcelain  sleeve :  then, 

w        log  (£)  X  4.5  log 

=  4.5 


Ep  ,      /R\  .      /1500\ 

log  (F)         log  (375) 

which  shows  the  voltage  across  the  small  air  space  to  be  greater 
than  that  across  the  greater  thickness  of  the  porcelain  bushing. 


INSULATION  OF  OVERHEAD  LINES  149 

The  voltage  gradient  which  will  cause  corona  or  brush  discharge 
on  the  surface  of  a  rod  of  radius  r,  according  to  Mr.  Peek,1  is 

Gv  —  31  (  1  H '~/= )  where  r  is  in  centimeters.     In  this  example, 

with  r  =  0.25  in.  we  find  the  value  of  Gv  to  be  42.7  k.v.  per  centi- 
meter. Then,  by  formula  (61)  the  maximum  permissible  value 
of  the  voltage  across  the  air  space  is 


Ea  =  42.7  X  0.25  X  2.54  X  loge  (^~)  =  H  k.v. 

The   total  permissible  pressure  across  the  bushing  is  therefore 
11   X    (l  +  i~3i5~)   =  19-37  k-v-  of  which  the  virtual  value  is 

19.36 

—/=-•=  13.67  k.v. 

Let  us  now  consider  the  stress  at  the  surface  of  the  conductor 
if  the  porcelain  sleeve  is  entirely  removed,  leaving  air  as  the 
only  dielectric  between  the  rod  and  the  outer  metal  cylinder. 
By  formula  (61), 

max.  value  of  E  =  42:7  X  0.25  X  2.54  X  loge  (Q^)  =  48.6  k.v. 

of  which  the  virtual  value  is  48.6/\/2  =  34.4  k.v. 

Thus,  if  the  porcelain  bushing  is  entirely  removed,  and  re- 
placed by  air,  the  voltage  (in  this  example)  may  be  increased  from 
13.67  to  34.4  k.v.  before  corona  will  form  at  the  surface  of  the 
rod.  Two  remedies  for  the  trouble  resulting  from  the  original 
design  suggest  themselves.  We  might  (1)  coat  the  inside  of  the 
porcelain  bushing  with  tin-foil  and  connect  this  electrically  with 
the  high-tension  rod,  or  (2)  we  might  fill  the  intervening  air 
space  with  oil  or  some  solid  insulating  compound  of  higher  spe- 
cific inductive  capacity  than  air. 

(1)  This  method  is  equivalent  to  making  the  high-tension  con- 
ductor of  radius  0.375.  Let  us  assume  the  total  voltage  to  be 
34.4  k.v.  as  calculated  for  the  arrangement  with  air  space  only. 
The  maximum  potential  gradient,  by  formula  (61),  will  then  be, 

Gmax.  =  -       34.4  X  1.41 ^^  =  36  g  k  y  per  centimeter 

0.375  X  2.54  X  logc  (Q^) 

1  "Dielectric  Phenomena,"  by  F.  W.  Peek,  Jr.,  McGraw-Hill  Book  Co. 


150  ELECTRIC  POWER  TRANSMISSION 

which  is  less  than  with  the  smaller  diameter  of  rod,  notwithstand- 
ing the  reduced  thickness  of  the  dielectric. 

(2)  If  we  assume,  for  simplicity  in  the  calculation,  that  the 
filling  compound  has  the  same  dielectric  constant  as  the  porcelain 
(k  =  4.5),  we  may  write, 

34  4  X  1  41 

(jmax.  =  -  '•          -  —  '-  -  =-=  —  =  42.7  k.v.  per  centimeter 


0.25  X  2.54  X  log€ 

which  is,  of  course,  the  same  as  when  the  entire  space  is  filled 
with  air;  but  there  will  be  no  corona  formation  on  the  surface 
of  the  high-tension  conductor. 

The  disadvantage  of  small  diameters  for  high-tension  conductors 
is  well  brought  out  by  this  example.  Without  any  increase  in  the 
size  of  the  outer  (grounded)  cylinder,  an  increase  in  diameter 
ofr  the  conductor  from  %  in.  to  %  in-  has  reduced  the  voltage 
gradient  at  the  conductor  surface  (where  its  value  is  a  maxi- 
mum) from  42.7  k.v.  to  36.8  k.v.  per  centimeter,  and  this  not- 
withstanding the  fact  that  the  distance  from  metal  to  metal, 
is  actually  less  in  the  case  of  the  lower  figure.  There  is  obviously 
a  limit  —  which  is  easily  calculated  —  beyond  which  any  increase  in 
the  diameter  of  the  inner  rod  would  lead  to  break-down  on  a 
lower  voltage. 

77.  Condenser  Type  of  Bushing.  —  By  separating  thin  concentric 
layers  of  insulating  material  by  tubes  of  tin-foil  or  other  metal, 
and  so  proportioning  the  lengths  of  the  tin-foil  tubes  that  the  areas 
remain  the  same,  notwithstanding  the  variations  in  diameter,  it  is 
possible  to  design  a  bushing  which  virtually  consists  of  a  number 
of  condensers  of  equal  capacity  all  connected  in  series.  In  this 
manner  the  potential  gradient  can  be  made  uniform  throughout 
the  thickness  of  the  insulating  bushing.  Commercial  entering 
bushings  and  transformer  terminals  have  for  some  time  past  been 
built  on  this  principle.  The  design  of  such  terminals  is  not  quite 
so  simple  a  matter  as  this  brief  reference  to  the  main  principle 
involved  might  suggest,  but  the  commercial  insulators  of  this 
class  have,  on  the  whole,  given  good  service. 

The  tendency  to  equalize  the  potential  gradient  throughout 
the  concentric  layers  of  insulating  material,  leads  to  a  bushing  of 
smaller  overall  diameter  than  when  no  attempt  is  made  to  equal- 
ize the  stresses.  Thus,  if  the  figure  of  36.8  k.v.  per  centimeter 
as  calculated  in  the  numerical  example  of  the  preceding  article 


INSULATION  OF  OVERHEAD  LINES  151 

is  assumd  to  be  the  maximum  working  stress  of  a  solid  porcelain 
bushing  of  inside  radius  %  in.  and  outside  radius  1^  in.,  a  bush- 
ing of  the  condenser  type  designed  for  the  same  maximum  stress 
and  suitable  for  the  same  total  voltage,  would  have  an  outside 
radius  of  only 

3       34.4  X  1.41 
R  =  8  +  36.8  X  2.54  =  °'893  m« 

This  calculation  does  not  take  account  of  the  thickness  of  the 
dividing  layers  of  tin-foil,  or  of  the  fact  that  perfect  equalization 
of  the  stress  is  not  obtainable  in  a  practical  design  of  bushing. 

Before  considering  such  matters  as  factors  of  safety,  spacing 
between  wires,  and  probable  limits  of  pressure  on  power  transmis- 
sion lines,  it  will  be  well  to  review  briefly  what  is  known  about 
brush'  discharges  and  corona  formation,  which  are  liable  to 
become  important  factors  when  transmitting  energy  at  the 
higher  voltages. 

78.  Formation  of  Corona,  and  Accompanying  Losses  of 
Power. — When  the  pressure  on  an  overhead  transmision  system 
exceeds  a  certain  critical  value  depending  upon  the  spacing  and 
diameter  of  the  wires,  there  will  appear  on  the  surface  of  the  con- 
ductors a  halo-like  glow  to  which  the  name  "corona"  has  been 
given.  Apart  from  this  luminous  effect,  the  appearance  of  the 
corona  is  accompanied  by  a  certain  loss  of  power  proportional  to 
the  frequency  and  the  square  of  the  amount  by  which  the  pres- 
sure between  conductors  exceeds  a  certain  value  known  as  the 
disruptive  critical  voltage.  If  the  distance  between  outgoing  and 
return  conductors  is  comparatively  small  (less  than  fifteen  times 
the  diameter  of  the  wire)  there  will  be  a  spark-over  when  the  dis- 
ruptive critical  voltage  is  reached;  but  with  the  greater  separation 
such  as  occurs  on  practical  high-tension  transmission  lines,  the 
effect  of  the  high  potential  at  the  conductor  surface  is  to  break 
down  the  resistance  of  the  air  in  the  immediate  neighborhood  of 
the  conductor  surface.  A  luminous  cylindrical  coating  of  air, 
acting  as  a  conductor  of  electricity,  is  thus  formed,  the  di- 
ameter of  which  will  depend  on  the  amount  by  which  the  actual 
value  of  the  applied  potential  difference  between  wires  exceeds 
the  disruptive  critical  value  of  the  potential  difference.  The 
result  is  equivalent  to  an  increase  of  the  diameter  of  the  conduct- 
ors, thus  raising  the  value  of  the  voltage  necessary  to  break  down 
new  concentric  layers  of  surrounding  air,  until  it  is  approximately 


152  ELECTRIC  POWER  TRANSMISSION 

equal  to  the  voltage  impressed  on  the  wires.  During  the  last 
few  years  much  light  has  been  thrown  on  the  formation  and  effects 
of  the  corona.  Among  the  earlier  workers  in  this  field  were  C.  F. 
Scott,  Harris  J.  Ryan,  J.  J.  Thomson,  H.  B.  Smith,  Signer  Jona, 
Lamar  Lindon,  E.  A.  Watson,  and,  among  the  later  investi- 
gators, J.  B.  Whitehead1  and  F.  W.  Peek,  Jr.2 

Suppose  a  cylindrical  wire  of  radius  r  is  surrounded  by  a  con- 
centric metal  cylinder  of  internal  radius  R  and  that  visible  corona 
starts  when  the  observed  voltage  is  Ev,  then  the  maximum  poten- 
tial gradient  at  the  surface  of  the  wire,  by  the  previously  developed 
formula  (61)  is 

(62) 


(D 


The  formula  for  parallel  wires  is 

Ev  (63) 


•*  $ 


where  d  is  the  distance  between  centers  of  wires. 

The  value  of  Gv  (the  apparent  strength  of  air)  is  found  to  be 
independent  of  the  spacing  between  wires,  but  it  is  not  found  to 
be  independent  of  their  diameter.  The  apparent  strength  of  air 
is  greater  at  the  surface  of  small  than  of  large  wires.  Mr.  Peek 
has  found  that,  at  a  distance  from  the  surface  of  any  cylindrical 
wire  equal  to  0.301  \/r  cm.,  the  breakdown  gradient  of  all  sizes 
of  wire  is  the  same,  namely  about  30  k.v.  per  centimeter.  This 
leads  to  the  formulas  which  will  be  found  at  the  end  of  this 
article. 

The  losses  which  occur  through  corona  are  not  different  in  kind 
from  I2R  losses;  but  since  they  occur  whenever  a  visual  corona 
appears,  and  may  reach  high  values  if  the  line  voltage  appreciably 
exceeds  the  voltage  at  which  corona  is  first  formed,  the  importance 
of  designing  high  tension  lines  so  as  to  avoid  excessive  corona 
formation  is  evident. 

The  more  important  features  and  effects  of  the  corona  of  in- 
terest to  the  practical  engineer  may  be  summarized  as  follows: 

1.  The  loss  due  to  leakage  of  current  from  the  conductor  into 

1Proc.  A.  I.  E.  E.,  vol.  xxix,  p.  1059  (1910),  and  later  contributions. 

zProc.  A.  I.  E.  E.,  vol.  xxxi,  p.  1085  (June,  1912).  Also  Journal  Franklin 
Institute,  Dec.,  1913;  and  Book  "Dielectric  Phenomena,"  McGraw-Hill 
Book  Co.,  1915. 


INSULATION  OF  OVERHEAD  LINES  153 

the  surrounding  air  is  practically  negligible  for  pressure  values 
below  the  disruptive  critical  voltage;  no  account  need  be  taken 
of  such  leakage  on  alternating-current  circuits  operated  at  pres- 
sures below  44,000  volts,  unless  the  wires  are  at  high  altitudes. 
On  80,000  volts,  however,  the  loss  may  be  appreciable,  and  a 
visible  corona  may  even  be  formed  if  the  wires  are  small  in 
diameter. 

2.  The  current  passing  from  the  wires  into  the  air  on  an  alter- 
nating system  is  an  energy  current  in  phase  with  the  pressure. 

3.  On    alternating-current    systems,  the  critical  break-down 
voltage  will  depend  upon  the  maximum  value  of  the  e.m.f.  wave, 
and  therefore  on  the  "form  factor." 

4.  The  break-down  voltage — or,  more  properly,  the  disruptive 
critical  voltage — is  determined  by  the  potential  gradient  at  the 
conductor  surface;  it  is  therefore  dependent  upon  the  diameter 
and  spacing  of  the  wires;  being  higher  with  the  larger  diameters 
and  spacings;  it  is  also  dependent  upon  the  density  of  the  air, 
and  therefore  on  the  temperature  and  barometric  pressure. 

5.  The  loss  of  power  due  to  corona  formation  is  approximately 
proportional  to  the  frequency   (within  the  usual   commercial 
range),  and  to  the  square  of  the  excess  of  line  voltage  over  the 
disruptive  critical  voltage. 

6.  The  disruptive  critical  voltage  is  the  voltage  at  which  the 
disruptive  voltage  gradient  of  the  air  is  reached  at  the  conductor 
surface.     It  is  highest  when  the  conductor  surface  is  smooth 
and  quite  clean.     It  is  lowered  by  roughness  or  dirt  on  the 
conductors,  also  by  smoke  and  fog  in  the  atmosphere,  sleet  on 
wires  and  falling  sleet,  rain  and  snow  storms,  and  low  barometric 
pressure.     All  these  causes  tend,  therefore,  to  increase  the  corona 
losses. 

7.  The  visual  corona  occurs  only  at  a  pressure  above  the  dis- 
ruptive critical  voltage  and  is  an  indication  that  there  is  loss  of 
power  in  the  air. 

When  considering  the  effects  of  the  corona  formation  on  over- 
head wires,  it  is  convenient,  as  in  the  case  of  the  majority  of 
electrical  problems  connected  with  transmission  lines,  to  consider 
each  wire  separately  in  relation  to  the  neutral  plane  or  line. 
Since  the  formation  of  the  corona  depends  upon  the  electric  stress 
at  the  surface  of  the  conductor,  it  is  the  potential  gradient  in  the 
immediate  neighborhood  of  the  wire  which,  as  previously  men- 
tioned, is  the  determining  factor  in  corona  formation.  The  dis- 


154  ELECTRIC  POWER  TRANSMISSION 

ruptive  critical  voltage  for  any  particular  wire  under  specified 
atmospheric  conditions  will,  as  previously  mentioned,  depend 
upon  the  diameter  of  the  wire  and  the  distance  of  the  wire  or  wires 
forming  the  return  conductor;  also  upon  the  density  of  the  air 
and,  to  some  extent,  upon  the  surface  condition  of  the  wire. 
Mr.  Peek's  formula  is: 

E0  =  21.lm0r8  log,  -  k.v.   to  neutral  (virtual  value)    (64) 

in  which  r  =  radius  of  conductor  in  centimeters, 
d  =  distance  between  centers  of  the  outgoing  and  return  (par- 

allel) conductors,  in  centimeters, 
m0=  a  factor  depending  upon  the  surface  condition  of  the  con- 

ductor, 

=  1  for  polished  wires, 

=  0.98  to  0.93  for  roughened  or  weathered  wires, 
=  0.87  to  0.83  for  stranded  cables  (average  =  0.85), 
5  =  a  factor  depending  on  the  air  density, 
3.926 
273  +  t 

in  which  b  is  the  barometric  pressure  in  centimeters  of  mercury, 
and  t  is  the  temperature  in  degrees  Centigrade. 

The  luminosity,  or  visible  halo  of  light  surrounding  the  con- 
ductor, does  not  occur  until  a  higher  pressure  has  been  reached, 
the  increase  over  the  critical  disruptive  voltage  being  dependent 
upon  the  diameter  of  the  conductor.  Mr.  Peek's  formula  for  the 
visual  critical  voltage  (kilovolts  to  neutral)  is: 

Ev  =  21.lm.r5    l  + 


r5  (l  +  ^=\  log,  j[  (65) 


where  the  surface  factor  mv  has  the  same  value  as  m0  for  wires, 
and  may  be  taken  at  0.82  for  a  decided  visible  corona  on  seven- 
strand  cables.  The  notation  is  otherwise  as  above. 

The  formula  for  loss  of  power  in  fair  weather,  in  kilowatts 
per  kilometer  of  single  wire,  as  given  by  Mr.  Peek,  is: 

P  =  ^  x  (f  +  25)  X-Jj  X  (En  -  E0Y  X  1(T5       (66) 

where  /  is  the  frequency  in  cycles  per  second,  and  En  is  the  actual 
(r.m.s.)  pressure  between  wire  and  neutral,  expressed  in  kilo- 
volts.  The  approximate  loss  under  storm  conditions  is  obtained 


INSULATION  OF  OVERHEAD  LINES  155 

by  taking  E0  as  80  per  cent,  of  its  (virtual)  value  as  calculated 
by  formula  (64). 

The  transmission  line  engineer  will  usually  prefer  formulas 
with  inch  units  and  common  logarithms;  and  the  following  may 
be  used: 


Eo  =  123  m0r8  Iog10          k.v.  to  neutral  (67) 

Ev  =  I23mvr5  (l  +  ^4§fyogw/^  k.v.  to  neutral        (68) 


r 

p  =       -  (/  +  25)          (En  -  E0Y  10-5  kw.  per  mile  of 

single  conductor     (69) 

The  air  density  factor  can,  if  desired,  be  calculated  by  the 
formula 

=     17.96 
459  +  t 

where  b  =  barometric  pressure  in  inches  of  mercury,  and  t  = 
temperature  in  degrees  Fahrenheit.  (Note  that  when  6  =  29.9 
and  t  =  77  degrees,  5=1.) 

As  a  guide  in  estimating  the  average  pressure  at  high  altitudes, 
the  following  figures  may  be  used  : 

Elevation,    sea  level,  b  =  29.9 

2,000  ft.,  6  =  27.6 

4,000  ft.,  b  =  25.6 

6,000  ft.,  b  =  23.7 

8,000  ft.,  &  =  22.0 
10,000  ft.,  6  =  20.4 
12,000  ft.,  6  =  18.9 

As  a  practical  example  of  corona  losses,  consider  a  100  mile, 
three-phase,  110,000  volt,  60-cycle  transmission,  using  No.  1 
seven-strand  conductors  spaced  6  ft.  apart.  Calculate  the 
approximate  fine-weather  corona  loss  if  the  air  density  factor  is 
unity  (5  =  1). 

By  formula  (67) 

Eo  =  123  X  0.85  X  0.165  X  log(V^)  =  45.6  k.v. 

XU.lOO/ 

By  formula  (69) 


=  5.1  kw.  per  mile  of  single  conductor. 


156  ELECTRIC  POWER  TRANSMISSION 

The  total  loss  to  be  expected  under  fair  weather  conditions  is 
therefore  5.1  X  3  X  100  =  1530  kw. 

The  assumption  is  here  made  that  the  line  voltage  (and  there- 
fore En)  remains  constant  over  the  entire  length  of  100  miles. 

79.  Corona  Considered  as  "  Safety  Valve"  for  Relief  of  High- 
frequency  Surges  or  Over -voltage  Due  to  Any  Cause. — The 
loss,  as  calculated  in  the  above  example,  is  not  small;  and  since 
it  is  proportional  to  the  square  of  the  excess  of  pressure  over  the 
disruptive  critical  voltage,  a  small  increase  of  pressure  will  lead 
to  an  enormously  increased  dissipation  of  energy  in  the  air. 
Thus,  if  the  pressure  of  110  kv.  in  the  above  example  be  supposed 
to  increase  only  10  per  cent,  the  total  dissipation  of  power, 
instead  of  being  1530  kw.,  would  be  2800  kw.     It  has  indeed  been 
stated  that  on  the  110,000-volt  system  of  the  Grand  Rapids- 
Muskegon.  Power  Co.,   the  line  loss  due  to  corona  discharge 
actually  increases  100  per  cent,  for  a  10  per  cent,  rise  in  pressure. 

This  property  of  the  corona  suggests  the  possibility  of  working 
high-voltage  transmission  lines  at  a  normal  pressure  in  the  neigh- 
borhood of  the  critical  disruptive  voltage  where  the  loss  would  be 
inappreciable.  An  extra-high-voltage  discharge,  due  either  to 
atmospheric  lightning,  or  to  internal  causes,  would  then  be  largely 
dissipated  in  the  corona  itself.  This  may,  to  some  extent, 
account  for  the  fact  that  fewer  lightning  troubles  are  experienced 
on  the  very  high  voltage  transmissions  than  on  the  lower  voltage 
lines.  The  insulation  of  the  conductors  being  such  as  to  with- 
stand, without  breakdown,  pressures  considerably  in  excess  of 
the  disruptive  critical  voltage  of  the  corona,  a  large  amount  of 
oscillating  energy  can  be  dissipated  in  the  air  before  the  voltage 
rises  to  such  a  value  as  to  pierce  or  shatter  insulators  or  damage 
apparatus  connected  to  the  line.  On  the  other  hand,  too  much 
reliance  should  not  be  placed  on  the  corona  as  a  means  of  dissi- 
pating large  amounts  of  suddenly  impressed  energy;  because 
lightning  and  similar  disturbances,  being  to  a  great  extent  local, 
must  discharge  their  power  locally,  and  the  corona  losses  over  a 
short  section  of  the  transmission  line  cannot  under  any  circum- 
stances be  very  great. 

80.  Spacing  of  Overhead  Conductors. — It  is  difficult  to  lay 
down  rules  for  the  proper  spacing  of  overhead  conductors.     The 
question  has  been  settled  in  the  past  by  the  individual  engineer 
who  has  usually  striven  to  be  "on  the  safe  side"  in  the  matter 
of  possible  discharges  between  wires  under  abnormal  conditions 


INSULATION  OF  OVERHEAD  LINES 


157 


such  as  strong  and  variable  winds.  The  result  is  that  great 
differences  are  to  be  found  in  the  wire  spacings  in  different  coun- 
tries or  on  different  transmission  systems  in  the  same  country. 
The  spacing  of  the  conductors  should  be  determined  by  consid- 
erations partly  electrical  and  partly  mechanical.  With  the  longer 
spans,  the  spacing  should  be  greater  than  with  short  spans, 


10   20   30   40   50   60   70   80   90   100  110  120  130  140 

Line  Pressure  (between  Wires.)  -.Kilov.olts 
FIG.  62. — Approximate  separation  of  overhead  conductors. 

apart  from  voltage  considerations.  The  material  and  diameter 
of  the  conductors  should  also  betaken  into  account  when  deciding 
upon  the  spacing,  because  a  small  wire — especially  if  of  aluminum 
— having  a  small  weight  relatively  to  the  area  presented  to  a 
cross  wind,  will  swing  out  of  the  vertical  plane  farther  than  a 
conductor  of  large  cross-section.  Usually  wires  will  swing 
synchronously  in  a  wind;  but  with  long  spans  and  small  wires, 


158  ELECTRIC  POWER  TRANSMISSION 

there  is  always  the  possibility  of  the  wires  swinging  non-syn- 
chronously,  and  the  size  of  wire,  together  with  the  maximum  sag 
at  center  of  span,  are  factors  which  should  be  taken  into  account 
in  determining  the  distance  apart  at  which  they  shall  be  strung. 
A  horizontal  separation  equal  to  something  between  one  and 
one  and  a  half  times  the  sag  at  the  temperature  corresponding 
to  the  season  of  highest  wind  velocities  should  be  sufficient  to 
prevent  wires  swinging  within  sparking  distance  of  each  other; 
the  closer  spacing  being  used  with  copper  conductors  of  large 
diameter. 

The  curves  of  Fig.  62  will  be  found  to  give  spacings  generally 
in  accordance  with  present-day  practice.  These  figures  may  be 
used  as  a  guide  in  arriving  at  a  suitable  value  for  the  horizontal 
spacings.  The  vertical  spacing  may  be  less;  but  it  is  usually 
undesirable  to  suspend  wires  in  the  same  vertical  plane,  especially 
in  locations  where  sleet  and  ice  deposits  are  likely  to  occur. 

Distance  Between  Conductor  and  Pole  or  Tower. — The  following 
clearances  are  recommended: 


Line  pressure,  k.v. 


10  (and  under) 

9 

15 

10 

22 

11 

35 

14 

44 

17 

66 

00 

24 

oo 

OO 

110 

Kfi 

36 

In  the  case  of  suspension  type  insulators  it  is  well  to  arrange 
for  the  clearance,  even  under  conditions  of  greatest  deflection 
caused  by  high  winds,  to  be  not  less  than  the  sparking  distance 
over  the  string  of  insulator  units. 

81.  Practical  Limitations  of  Overhead  Transmission-line 
Voltages. — From  the  foregoing  review  of  the  insulation  prob- 
lems to  be  met  with  on  long-distance  overhead  transmissions, 
it  will  be  clear  that  manufacturers  are  now  in  a  position  to  pro- 
vide insulation  amply  sufficient  for  present  requirements.  Power 
is  actually  being  transmitted  at  150,000  volts.  The  lines  of 
the  Au  Sable  Electric  Co.  in  Michigan,  transmitting  power  at 


INSULATION  OF  OVERHEAD  LINES  159 

140,000  volts,  consist  of  stranded  copper  conductors  about  %-in. 
diameter  on  500-ft.  spans,  with  a  sag  allowance  of  approximately 
12  ft.  The  shortest  distance  between  conductors  is  12  ft.,  this 
being  the  vertical  height  between  the  two  conductors  on  one  side 
of  the  steel  supporting  towers.  There  is  practically  no  visible 
corona,  but  a  buzz  or  hum,  due  no  doubt  to  brush  discharge,  can 
be  heard  in  the  neighborhood  of  the  transmission  lines. 

Although  there  are  no  insurmountable  difficulties  in  providing 
ample  insulation  for  these  high  voltages,  it  is  the  engineer's 
business  to  provide  such  insulation  as  will  be  justified  by  economic 
considerations.  It  is  also  his  business  to  determine  the  voltage  of 
transmission  on  the  same  basis,  and  resist  the  temptation  to 
experiment  in  voltages  higher  than  may  be  justified  by  commer- 
cial considerations. 

It  is  well  to  bear  in  mind  that  the  economical  transmission 
voltage  depends  not  only  on  the  length  of  the  line,  but  also  on 
the  amount  of  power  to  be  transmitted;  and  although  a  200,000- 
volt  transmission  offers  no  serious  engineering  difficulties,  the 
conditions  under  which  a  transmission  at  so  high  a  voltage 
would  be  a  commercial  success,  are  very  seldom  found. 

82.  Factors  of  Safety :  Rating  and  Testing  of  Line  Insulators. — 
When  selecting  insulators  and  deciding  upon  the  spacing  and 
arrangement  of  conductors  suitable  for  a  given  voltage,  the  factor 
of  safety  to  cover  abnormal  pressure-rises  is  a  matter  of  great 
importance,  since  it  is  obviously  bad  engineering  to  provide 
insulation  in  excess  of  what  experience  has  shown  to  be  a  reason- 
able safeguard  against  interruption  of  service.  Generally 
speaking,  the  insulators  should,  when  dry,  withstand  a  pressure 
test  of  2^2  to  3  times  the  working  pressure  to  ground,  applied 
for  five  minutes,  and  a  wet  test  of  not  less  than  twice  the  working 
pressure.  This  would  sometimes  be  considered  too  small  a 
margin  of  safety;  but  the  ratio  between  test  pressure  and  working 
pressure  will  depend  upon  whether  the  line  voltage  is  high  or  low. 
The  following  safety  factors,  representing  ratio  between  wet-test 
pressure  and  working  pressure  are  generally  in  accordance  with 
usual  practice;  but  the  engineer  should  use  his  judgment  in  a 
matter  of  this  sort.  It  is  clear  that,  on  the  coast,  where  gales 
and  salt  sea  mists  are  prevalent,  the  factor  of  safety  should  be 
rather  higher  than  in  a  district  where  the  climatic  conditions  are 
more  favorable.  The  effects  of  high  altitude — to  be  referred 
to  later — must  also  be  taken  into  account, 


160 


ELECTRIC  POWER  TRANSMISSION 


Working  pressure 
(voltage  between  line  wires) 

Safety  factor 
(wet  test) 

20000  
40000   

2.5 
2.2 

80000....  

2.0 

Above  80000  volts 

1  8  to  2 

As  the  wet  or  "rain"  test  will  give  different  results,  depending 
on  the  method  of  conducting  the  tests,  there  should  be  a  clear 
understanding  between  the  purchaser  and  manufacturer  on  this 
point.  A  very  common  specification  is  that  the  spray  shall  be 
directed  at  an  angle  of  45  degrees,  under  a  pressure  of  40  Ib. 
per  square  inch  at  the  nozzles;  the  flow  being  regulated  to  give 
a  precipitation  of  1  in.  in  5  minutes.  The  method  of  attaching 
test  wire  and  ground  connection  to  the  insulator  should  also  be 
clearly  defined.  The  test  pressure  is  usually  measured  by  means 
of  a  spark  gap;  and  the  alternating  e.m.f.  used  should  conform  as 
nearly  as  possible  to  the  sine  wave . 

The  striking  distances  in  air  between  No.  3  sharp  needles,  as 
given  by  the  Locke  Insulator  Mfg.  Co.,  are  as  follows: 


Kilovolts 


Kilovolts 


20  |  

1.00 

200  

20.50 

40  

2.45 

225  

23.05 

60  

4.65 

250  

25.60 

80  

7.10 

275  

28.30 

100  

9.60 

300  

31.00 

125  

12.25 

350 

36.10 

150  

15.00 

400  

41.20 

175  

17.80 

It  will  be  observed  that  for  pressures  above  100,000  volts  the 
gap  between  needle  points  is  approximately  1  in.  per  10,000  volts. 
The  pressures  referred  to  in  the  table  are  the  virtual  or  root- 
mean-square  values  of  the  test  pressure,  on  the  sine  wave 
assumption. 

Effect  of  Altitude. — The  insulators  on  a  transmission  line 
erected  at  high  altitudes  will  flash  over  with  a  lower  voltage  than 
if  the  line  were  erected  at  sea  level.  The  reason  for  this  is  the 
reduced  air  pressure  at  the  higher  elevation.  The  flash-over 
voltage  will  not  be  exactly  proportional  to  the  barometric  pres- 


INSULATION  OF  OVERHEAD  LINES  161 

sure  because  the  electrostatic  field  is  not  uniform,  but  depends 
upon  the  type  and  design  of  the  insulator.  Mr.  Peek  gives  some 
results  of  experimental  work  on  various  standard  designs  of 
insulator  in  his  book  on  Dielectric  Phenomena  previously  referred 
to;  but  since  the  departures  from  the  theoretical  relation  for 
uniform  dielectric  fields  is  very  small,  the  correction  for  altitude 
can  safely  be  made  by  assuming  the  flash-over  voltage  to  be 
directly  proportional  to  the  barometric  pressure. 

As  an  example:  suppose  the  flash-over  voltage  of  a  pin  type 
insulator  is  found  to  be  100  kv.  on  test  at  sea  level;  then,  if 
used  on  a  transmission  line  erected  at  an  elevation  of  8000  feet, 
it  would  be  liable  to  flash  over  with  a  pressure  of  only  100  X 
22 
299  =  73'^  ky*  (The  proportional  figures  for  air  density 

are  taken  from  the  table  on  page  155.)  It  is  thus  easy  to  de- 
cide upon  a  factor  of  safety  which  shall  make  proper  allowance 
for  the  elevation  at  which  the  insulator  may  have  to  operate. 

Rating  of  Insulators. — The  fact  that  conditions  of  tests  and 
factors  of  safety  may  have  to  be  modified  in  accordance  with 
the  conditions  under  which  the  insulator  will  be  expected  to 
operate,  suggests  the  subject  of  insulator  rating.  At  the  present 
time  this  question  is  far  from  being  settled  in  a  satisfactory 
manner,  and  it  would  seem  that  manufacturers  and  users  should 
get  together  with  a  view  to  deciding  upon  an  acceptable  basis 
which  would  permit  of  any  design  of  insulator  being  placed  in  a 
particular  class,  and  so  facilitate  the  comparison  between  tenders 
submitted  by  competing  firms  for  insulators  to  operate  under 
given  conditions.  At  first  sight  the  most  reasonable  method 
appears  to  be  to  adopt  some  arbitrary  rule  depending  upon  the 
dry  and  wet  flash-over  voltage  of  the  insulator:  for  instance, 
to  specify  that  for  the  higher  line  pressures  the  dry  flash-over 
should  be  three  times  the  line  voltage,  and  the  wet  flash-over 
twice  the  line  voltage.  This  would,  however,  probably  lead  to 
confusion  because  of  the  number  of  variable  factors  involved 
in  determining  the  dry  and  wet  flash-over.  It  is  possible — as 
suggested  to  the  writer  by  an  engineer  connected  with  one 
of  the  best  known  firms  manufacturing  porcelain  insulators — 
that  a  method  of  rating  depending  not  upon  tests,  but  upon 
design  dimensions,  such  as  the  overall  height  and  the  minimum 
arcing  distances  (both  wet  and  dry),  might  prove  a  useful 
basis  of  classification. 


162  ELECTRIC  POWER  TRANSMISSION 

Detecting  Faulty  Insulators  while  Line  is  in  Operation. — Another 
question  which  is  now  receiving  attention,  but  is  not  settled 
or  standardized,  is  the  best  means  of  detecting  faults  in  insulators 
while  in  use.  It  is  only  of  recent  years  that  accurate  data  on  the 
"life"  of  high  tension  insulators  is  becoming  available;  and 
the  continued  action  of  alternate  heat  and  cold,  dryness  and 
dampness,  on  the  porcelain — or  rather  on  the  complete  assembly 
of  porcelain,  metal,  and  cement — is  found  to  necessitate  a  very 
large  percentage  of  replacements  after  a  line  has  been  in  operation 
many  years. 

The  causes  of  rapid  deterioration — especially  after  several 
years  of  service — are  being  investigated,  and  eliminated  as  far 
as  possible  in  the  later  designs;1  but  in  the  meanwhile,  lines 
that  have  been  in  use  for  a  considerable  time  are  giving  trouble 
in  the  matter  of  insulation,  involving  increasing  vigilance  and 
activity  on  the  part  of  the  operating  staff.  There  is  room  for 
improvement  in  the  methods  now  available  for  detecting  in- 
cipient faults  in  line  insulators  without  interrupting  the  supply  or 
disconnecting  insulators  from  the  live  wires.  One  method,  which 
involves  the  use  of  a  telephone,  and  is  said  to  give  reliable 
information  as  to  the  condition  of  the  insulators,  is  described  on 
page  821  of  Vol.  64  of  the  Electrical  World  (Oct.  24,  1914).  The 
reader  who  is  interested  in  this  matter  of  insulation  troubles 
should  also  refer  to  an  article  by  Prof.  H.  J.  Ryan  in  the  Journal 
of  Electricity  (San  Francisco),  Feb.  27,  1915;  and  to  the  article 
"Testing  Insulators  to  Assure  Continuous  Service"  by  Professor 
R.  W.  Sorensen  in  the  Electrical  World  of  Sept.  1,  1917  (Vol. 
70,  p.  426). 

1  Refer  Trans.  A.  1.  E.  E.,  vol.  xxxiii  (1914),  pp.  Ill  and  119  (J.  A.  Brun- 
dige),  and  p.  1731  (A.  O.  Austin).  Vol.  xxxiv  (1915),  p.  465  (E.  E.  F. 
Creighton).  Vol.  xxxvi  (1917),  p.  527  (W.  D.  Peaslee),  p.  535  (J.  A. 
Brundige),  and  p.  545  (A.  O.  Austin). 


CHAPTER  VI 

PROTECTION   AGAINST  LIGHTNING— TRANSIENT 
PHENOMENA 

83.  Theoretical  Considerations. — Before  taking  up  the  matter 
of  lightning  disturbances  and  the  means  adopted  for  minimizing 
their  destructive  effects  on  line  insulation  and  station  apparatus, 
the  causes  leading  to  surges,  oscillations,  and  travelling  or  stand- 
ing waves,  will  be  very  briefly  discussed.  No  attempt  will 
be  made  to  deal  thoroughly  with  this  somewhat  difficult  subject 
which  has  lately  come  into  prominence  because  of  the  increasing 
distances  to  which  energy  is  being  transmitted  electrically. 
Nothing  will  be  included  here  beyond  the  elementary  considera- 
tions with  which  the  engineer  engaged  on  the  design  of  high- 
tension  long-distance  transmission  lines  should  be  familiar. 
For  a  complete  study  of  the  principles  underlying  transient 
phenomena — energy  surges  and  oscillations,  and  the  peculiarities 
of  hypothetical  "quarter  wave  length"  transmission  lines — 
the  reader  is  referred  to  authorities  such  as  Dr.  Steinmetz1 
and  the  writings,  by  various  authors,  that  have  appeared  recently 
in  the  technical  Journals  and  the  publications  of  the  Scientific 
and  Engineering  Societies. 

The  relation  existing  between  the  voltage  #nd  the  current  of  any 
transient  electrical  disturbance  occurring  in  a  circuit  depends 
upon  the  relation  between  the  magnetic  flux-linkages  per  unit 
current — or  the  inductance — and  the  permittance  or  electro- 
static capacity. 

Consider  a  circuit  in  which  there  is  alternating  or  oscillating 
energy  which  is  not  utilized  by  any  form  of  receiving  apparatus, 
and  is  not  dissipated  in  the  form  of  heat  through  the  ohmic 
resistance  of  the  conductors,  or  through  "dielectric  hysteresis" 
or  corona:  it  is  obvious  that,  at  the  instant  when  the  current 
wave  passes  through  zero  value,  the  whole  of  the  energy  must  be 
stored  in  the  electrostatic  field,  and  similarly,  at  the  instant  when 
the  pressure  wave  passes  through  zero  value,  the  whole  of  the 
oscillating  energy  must  be  stored  in  the  electromagnetic  field. 

1  "Transient  Electric  Phenomena  and  Oscillations." 
163 


164  ELECTRIC  POWER  TRANSMISSION 

Moreover,  so  long  as  the  interchange  of  energy  from  one  form  to 
the  other  continues  without  diminution  of  amount,  these  two 
quantities  must  be  exactly  equal.  This  conception  of  the  oscil- 
lations of  energy  in  a  circuit  having  negligible  resistance,  but 
appreciable  inductance  and  capacity,  is  fundamental,  and  we 
shall  examine  it  in  further  detail  with  a  view  to  arriving  at  a 
definite  relation  between  the  amplitudes  of  the  voltage  and  current 
waves. 

Energy  Stored  in  Magnetic  Field. — Since  the  engineer  usually 
prefers  to  think  of  volts  and  amperes,  the  product  of  which  repre- 
sents power  (watts)  or  the  rate  at  which  work  is  being  done,  we 
may  say  that  the  energy  stored  in  a  magnetic  field  during  a  short 
interval  of  time  dt  seconds  is  ei  X  dt  watt-seconds  or  joules.  In 
this  connection  the  voltage  e  is  the  e.m.f .  developed  in  a  conductor 
carrying  i  amperes  when  a  change  in  current  di  causes  a  change 
of  flux  d$  in  the  short  interval  of  time  dt.  Thus,  since  we  are 
considering  the  flux  linking  with  a  circuit  of  one  turn  (a  trans- 
mission line  conductor)  in  a  medium  (air)  of  constant  permeability, 
we  may  write, 

d<£      T  di 

e==~dt=Ldi 

whence 

Energy  stored  in  magnetic  field  during  the  |    _  T  di '       .        , 
interval  of  time  dt  }    "    L~dt  X 

=  Li  X  di 

and  since  the  current  grows  from  zero  to  its  maximum  value 
in  a  quarter  of  a  period,  we  have, 

Energy  stored  in  magnetic  field  1  _       j  •1^5Z^- 
during  one  quarter   period  J 

=  Yz  LJ2max.  joules  (70) 

It  is  easy  to  show  in  a  similar  manner  that  the  energy  stored 
in  the  dielectric  circuit  in  one  quarter  period  while  the  value 
of  e  grows  from  zero  to  jKmax.  is  %  CEzm&^  where  C  is  the  electro- 
static capacity  of  the  circuit,  or  portion  of  circuit  considered. 

Thus,  in  the  case  of  a  pure,  undamped,  oscillation,  when  no 
energy  is  supplied  from  the  outside  to  the  circuit,  or  by  the  cir- 
cuit to  the  outside,  it  follows  that 

L72  =  CEZ 


PROTECTION  AGAINST  LIGHTNING  165 


(7D 

where  L  is  expressed  in  Henrys,  and  C  in  farads. 

The  quantity  -»/^  is  thus  seen  to  be  of  the  nature  of  a  resistance 

or  impedance,  and  it  may  be  expressed  in  ohms.  It  is  generally 
called  the  natural  impedance  of  the  circuit;  but  the  expressions 
wave  impedance  (or  resistance)  and  surge  impedance  (or  resistance) 

are  also  used  to  denote  the  ratio  —        —  of  the  oscillating  energy. 

amperes 

In  the  case  of  an  overhead  transmission  line,  the  approximate 
value,  per  mile,  of  the  inductance  betwen  one  conductor  and 
neutral,  as  given  in  Article  48,  Chapter  IV  (page  88)  is 

External  Inductance,  L  =  0.000741  log  - 

and  the  approximate  formula  for  capacity,  as  given  in  Article  10 
of  Chapter  II,  is 

farads 


whence  the  surge  impedance  of  an  overhead  transmission  line  is 
approximately, 


g  =  138  log  -  ohms  (72) 

In  practical  overhead  work,  the  limiting  values  for  the  ratio 
-  will  probably  be  800  and  50;  which,  when  inserted  in  formula 

(71),  show  that  the  "natural  impedance"  of  an  overhead  trans- 
mission line  must  lie  between  400  and  230,  or,  to  be  well  on  the 
safe  side,  between  (say)  500  and  200  ohms. 

A  knowledge  of  this  quantity  renders  it  possible  to  determine 
the  maximum  value  of  any  surge  pressures  that  can  possibly 
occur  on  the  line  due  to  the  sudden  interruption  of  the  current. 
Thus,  if  the  " natural  impedance"  is  300  ohms,  and  the  instanta- 
neous value  of  the  current  at  the  crest  of  the  wave  is  200,  the  surge 
pressure,  however  suddenly  the  current  is  interrupted,  cannot 
possibly  exceed  200  X  300  =  60,000  volts;  because  this  is  the  maxi- 
mum value  of  the  pressure  wave  necessary  to  store  in  the  elec- 
tric field  the  whole  of  the  energy  stored  in  the  magnetic  field  at 


166  ELECTRIC  POWER  TRANSMISSION 

the  moment  when  the  current  was  interrupted.  It  is  safe  to  say 
that,  on  a  practical  transmission  line,  the  surge  pressure  is  never 
likely  to  exceed  200  times  the  current  in  amperes;  but,  with  heavy 
currents,  this  may  well  be  sufficient  to  break  down  insulation  and 
cause  considerable  damage  to  power  plant.  It  must  not  be  over- 
looked that  it  is  often  more  difficult  to  handle  heavy  currents 
at  comparatively  low  pressures  than  small  currents  at  the  very 
highest  pressures  yet  attempted.  When  the  current  is  large,  the 
opening  of  switch  or  fuse  on  full-load,  or  an  accident  causing  a 
break  in  the  circuit,  with  or  without  the  formation  of  an  arc 
across  the  gap,  may  lead  to  insulation  troubles  on  many  widely 
separated  parts  of  the  system;  but  on  a  high-pressure  system, 
even  if  the  current  were  as  large,  the  insulation  is  frequently  so 
good  that  it  will  withstand  without  injury  the  stress  imposed  on 
it  by  the  highest  possible  value  of  the  surge  pressure. 

In  underground  cables,  the  capacity  is  much  larger  relatively 
to  the  inductance  than  in  overhead  systems,  and  the  surge  im- 


pedance, ATV  has  then  a  smaller  value,  which  may  be  about 

one-tenth  of  the  value  for  overhead  lines;  but  the  transformers 
connected  to  transmission  systems  will  always  have  a  surge 
impedance  very  considerably  higher  than  that  of  the  line  itself. 

The  effect  of  the  ohmic  resistance  in  series  with  the  'inductive 
and  condensive  reactances  of  a  circuit,  is  to  damp  out  the  oscilla- 
tions by  dissipating  the  energy  in  the  form  of  I2R  losses.  With 
a  sufficiently  high  value  of  resistance  in  the  circuit,  surges  or 
oscillations  of  energy  cannot  take  place. 

For  the  condition  of  massed  resistance,  capacity,  and  induc- 
tance, the  critical  resistance  is 


R  = 

and  if  R  has  a  higher  value  than  this,  oscillations  of  energy  cannot 
occur.  The  case  of  a  transmission  line  with  distributed  capacity 
and  inductance  is  much  more  complicated,  and  the  mathematical 
analysis  is  very  difficult;  but  if  the  receiving  end  of  a  long  trans- 
mission line  of  negligible  resistance  is  closed  through  a  resistance 
(line  to  neutral)  of  value 


R  =  \fe  (73) 


PROTECTION  AGAINST  LIGHTNING  167 

there  will  be  no  oscillations  of  energy  resulting  from  a  sudden 
change  of  potential.  In  other  words,  a  surge  travelling  along 
the  line  will  be  completely  absorbed,  and  there  will  be  no  "re- 
flected" waves.  (The  proof  of  this  statement  will  be  given  in 
Article  86). 

If  R  has  a  value  smaller  than  that  given  by  formula  (73)  there 
will  be  " reflection"  of  some  part  of  the  impressed  energy,  and 
oscillations  will  occur;  but  these  will  gradually  decrease  in  ampli- 
tude according  to  the  logarithmic  law. 

84.  Frequency  of  Oscillations. — The  rate  at  which  the  oscil- 
lating energy  will  pass  back  and  forth  between  the  magnetic  and 
the  dielectric  fields  is  entirely  independent  of  the  frequency  of 
the  power  current  in  a  transmission  line.  By  formula  (71)  we 
have 


E  =  I^~  (74) 

but  since  /  may  be  considered  as  the  charging  current  of  a  conden- 
ser of  capacity  C  with  a  voltage  E  across  the  terminals,  we  can 
also  write 

E=2^fC  (75) 

which  is  obtained  from  formula  (36)  of  Article  54,  Chapter  IV. 
The  value  of  /  obtained  by  equating  (74)  and  (75)  is 

/  = j=.  periods  per  second  (76) 

27r\/LC 

which  is  the  periodicity  of  the  oscillations  when  the  inductance 
and  capacity  are  supposed  to  be  concentrated  at  a  given  point. 
When  capacity  and  inductance  are  distributed  as  on  -a  long-dis- 
tance power-transmission  line,  the  formula  for  the  frequency  of 
oscillation  as  developed  by  Dr.  Steimnetz  is 

(77) 


which  is  the  frequency  of  resonance  and  is  called  the  natural 
frequency  of  the  line.  It  should  be  noted  that  L  and  C,  as  in 
the  previous  formulas,  are  expressed  in  henrys  and  farads;  but 
since  we  are  dealing  with  a  product,  not  a  ratio,  of  the  two  quan- 
tities, it  is  the  inductance  per  mile  X  length  in  miles,  and  the 
capacity  per  mile  X  length  in  miles,  that  these  symbols  now  stand 
for. 


168  ELECTRIC  POWER  TRANSMISSION 

By  formula  (37)  of  Article  54,  Chapter  IV,  the  approximate 
value  of  the  product  CL  for  an  overhead  transmission  one  mile 
long  is  seen  to  be 

1 


CL 


34700  X  106 


which,  when  substituted  in  formula  (77),  gives  for  the  natural 
frequency  of  an  overhead  line, 


(78) 


where  L  is  the  distance  of  transmission,  or  length  of  a  single 
conductor,  in  miles. 

85.  Wave  Length.  —  The  rate  of  travel  of  an  electric  impulse 
along  an  overhead  wire  is  approximately  the  same  as  the  velocity 
of  light,  or  (say)  186,000  miles  per  second.  Thus,  if  we  imagine 


< 775  miles-  ->i 

3100  miles 


Fio.  63. — Diagram  showing  instantaneous  value  of  current  at  different  points 
on  a  long  transmission  line. 


an  alternating  e.m.f .  of  frequency  /  =  60  applied  to  the  ends  of 

186000 

a  circuit  of  length  —       —  =  3100   miles,    the  maximum  value 
oU 

of  the  current  wave  would  occur  at  the  receiving  end  of  the  line 
simultaneously  with  the  maximum  value  at  the  sending  end,  but 
it  would  be  the  crest  of  a  current  wave  which  had  left  the  sending 
end  exactly  3-1$  o  second  earlier.  Fig.  63  is  an  attempt  to  indi- 
cate the  travel  of  the  electrical  impulse  over  a  wire  of  great 
length.  The  ordinates  of  the  curve  show  the  value  of  the  current 
at  any  point  along  the  line  at  the  instant  when  the  impressed  sine 
wave  of  current  has  attained  its  maximum  value  at  the  sending 
end  of  the  line.  The  wave  length,  or  distance  covered  by  one 


PROTECTION  AGAINST  LIGHTNING  169 

1  86000 
complete  wave,  is  -  j  -  miles,  which  in  this  example  is  3100 

miles. 

A  quarter  wave  length  line  with  a  frequency  of  60  would  be 

775  miles  long;  it  would  have  the  peculiarity  that  the 

pressure  (or  current)  wave  would  have  zero  value  at  one  end  of 
the  line  at  the  same  instant  of  time  as  its  value  was  a  maximum 
at  the  other  end.  The  characteristics  of  such  a  line  are  quite 
different  from  those  of  an  ordinary  transmission  line,  and  al- 
though at  ordinary  frequencies  trouble  from  this  cause  is  not 
likely  to  result,  it  is  possible  to  get  the  resonance  effect  of  a 
quarter  wave  line  with  the  higher  harmonics  of  a  distorted  wave, 
even  on  practical  transmission  lines. 

The  natural  period  of  an  overhead  line,  as  given  by  formula 
(78)  is 

1       186000 


in  which  L  is  now  seen  to  be  the  length  in  miles  of  a  quarter  wave 
transmission  line,  although  it  was  not  previously  pointed  out  that 
the  constant  186,000  in  formula  (78)  actually  represents  the  veloc- 
ity of  light  (or  of  an  electric  impulse)  in  miles  per  second. 

Without  attempting  to  explain  or  analyze  the  properties 
peculiar  to  a  quarter  wave  length  transmission  line,  it  may  be 
said  that  these  are  largely  due  to  the  fact  of  the  quarter  wave 
displacement  providing  the  charging  current  for  the  line,  and  so 
leaving  the  generators  to  supply  the  load  and  losses.  The  induc- 
tive pressure  drop  and  the  charging  current  are,  in  effect,  wiped 
out  by  the  peculiar  overlapping  of  the  travelling  waves  of  energy. 
The  power  factor  of  a  line  specially  designed  to  make  use  of  this 
peculiarity  would,  therefore,  be  very  nearly  100  per  cent,  at  all 
loads,  and  the  regulation,  even  if  the  load  were  inductive,  might 
be  surprisingly  good.  With  distorted  waves,  and  complications 
due  to  limited  length  of  line,  branch  circuits,  and  other  causes, 
it  is  usually  desirable  to  avoid  the  conditions  of  resonance  in 
practice. 

As  an  example,  consider  a  line  200  miles  long  :  what  frequency 
will  cause  the  quarter  wave  effect?  This  is  the  frequency  which 
causes  the  conditions  at  the  sending  end  to  be  repeated  at  the 


170 


ELECTRIC  POWER  TRANSMISSION 


receiving  end  exactly  one  quarter  of  a  period  later,  and  by  formula 
(78)  we  have 

/  =  4X2QO  =  233  cycles  per  second 

which  is  the  lowest  frequency  at  which  free  oscillations  can  occur 
on  a  transmission  line  of  this  length.  It  corresponds  to  the  third 
harmonic  of  a  wave  of  fundamental  frequency  77.7,  which  should 
therefore  be  avoided;  but  either  60  or  25  cycles  will  be  satis- 
factory on  a  line  of  this  length. 

86.  Reflection  of  Travelling  Waves. — Imagine  a  three-phase 
transmission  line  arranged  as  shown  in  Fig.  64.  Here  G  is  a 
three-phase  generator  from  which  a  voltage  e  as  measured  be- 
tween line  (1)  and  ground  (or  neutral  point  of  star  connection)  is 
suddenly  impressed  for  a  very  short  space  of  time,  by  closing 
and  then  immediately  opening  the  three-pole  switch  S.  We  have 
here  the  case  of  a  "wave  pulse"  travelling  along  the  transmission 
line  with  the  velocity  of  light.  Let  us  consider  what  happens 
when  it  reaches  the  end  of  the  line,  for  the  three  conditions, 

C  d  (3) 


(2) 


FIG.  64. — Diagram  illustrating  "wave  pulse"    travelling  along  a  three-phase 
transmission  line. 

1.  Line  open  (/?=<») 

2.  Line  short-circuited  (R  =  0) 

3.  Line  closed  through  a  non-inductive  resistance  of  value  R 
(line  to  neutral). 

Suppose  that  the  operations  of  closing  and  opening  the  switch 
have  been  performed  so  quickly  that  we  have  a  rectangular  wave 
pulse  of  which  the  length  ab  depends  upon  the  time  during  which 
the  switch  was  closed.  The  current  i  flows  in  the  wire  between 
the  points  a  and  6,  but  neither  forward  of  6  nor  backward  of  a. 
It  carries  with  it  the  system  of  magnetic  and  dielectric  flux  lines 
in  the  space  comprised  between  the  planes  ac  and  bd;  the  complete 
energy  pulsation  being  supposed  to  move  away  from  G  with  the 
Velocity  of  light. 


PROTECTION  AGAINST  LIGHTNING  171 

Assuming  the  resistance  of  the  conductors  to  be  negligible,  the 
relation  between  i  and  e  is  given  by  formula  (7 1).1 

To  satisfy  Case  (1),  the  current  arriving  at  the  open  end  of 
the  line  must,  at  every  instant,  be  equal  to  the  current  leaving 
this  point;  and  since  R  =  oo  (i.e.,  since  the  circuit  is  open)  i  +  i' 
=  0;  the  symbol  i'  standing  for  the  " reflected"  current. 

In  order  to  reverse  the  current  and  start  the  wave  pulse  back 
toward  the  generator  end  of  the  line,  we  must  conceive  of  a 
"piling  up''  of  the  e.m.f.  at  the  open  end  of  the  line  sufficient 
to  force  the  reflected  current  i'  against  the  incoming  current  i. 
But  that  is  exactly  what  must  necessarily  occur  because  when  half 
of  the  wave  pulse  of  current  has  been  reflected,  the  resultant 
current  in  the  line  will  be  zero,  and  since  no  energy  has  been  lost, 
it  must  all  be  in  the  electrostatic  field. 

Apart  from  the  relation  i  +  *'  =  0,  at  the  open  end  of  the  Line, 
the  equations 


for  the  outgoing  wave,  and 


---  F 

ir    Vc 


for  the  reflected  wave,  must  be  satisfied;  whence  it  follows  that 
e  =  e'  when  i  =  —i'. 

In  other  words,  during  the  period  of  reversal  of  the  current 
wave,  at  all  portions  of  the  line  where  i  =  —  i',  the  voltage  must 
be  2e.  On  arrival  at  the  point  where  the  reversal  of  the  wave 
pulse  is  complete,  the  voltage  will  again  have  its  original  value 
of  e  volts. 

Case  2. — With  a  short-circuit  at  the  end  of  the  line,  complete 
reflection  of  the  wave  pulse  will  also  occur,  but  it  will  now  be 
the  voltage  instead  of  the  current  that  is  reversed.  Since  there 

1  This  relation  is  not  obviously  true  of  a  travelling  wave  or  impulse;  but 
in  order  to  avoid  devoting  a  disproportionate  amount  of  space  to  this  sub- 
ject, it  is  necessary  to  refer  the  reader  to  other  sources  of  information, 
should  he  find  this  brief  treatment  inadequate  to  his  needs.  The  most 
practical  and  the  clearest  explanation  of  the  peculiarities  of  travelling  waves, 
known  to  the  writer,  will  be' found  in  Franklin  and  MacNutt's  "Advanced 
Electricity  and  Magnetism."  These  authors  present  the  subject  in  a 
manner  that  will  satisfy  the  requirements  of  most  electrical  engineers  far 
better  than  the  many  highly  mathematical  writings  on  the  same  subject  by 
other  authors. 


172  ELECTRIC  POWER  TRANSMISSION 

is  a  short-circuit  at  the  end  of  the  line,  the  volts  must  be  zero 
when  the  wave  pulse  is  in  process  of  reversal,  and  in  order  to 
maintain  the  total  energy  constant  (a  necessary  condition)  the 
current  where  the  overlapping  occurs  must  be  2i.  When  reversal 
is  complete,  the  current  will  be  *  as  before  —  that  is  to  say, 
of  the  same  amount  and  direction;  but  the  voltage  e  will  be 
of  opposite  sign  because  the  wave  pulse  of  energy  is  now  travelling 
along  the  line  in  the  reverse  direction. 

Case  3.  —  Comparing  the  two  preceding  extreme  cases,  it  is 
evident  that  since  er  =  e  and  i'  =  —  i  when  #  =  oo,  while 
e'  =  —  e  and  i'  =  i  when  R  =  0,  there  must  be  a  particular 
intermediate  value  of  R  which  will  absorb  the  wave  pulse  and 
prevent  reflection. 


When  R  is  made  equal  to  ~-^>  reflection  cannot  occur  because 

both  voltage  and  current  waves  will  enter  the  resistance  un- 
changed. 

Using  symbols,  we  may  say  that  for  any  value  of  R,  if  i  is  the 
outgoing  current  and  i'  is  the  reflected  current,1  the  balance 
which  enters  the  resistance  is  ir  =  (i  +  i'}.  The  volts  at  the 
end  of  the  line  must  therefore  be  (e  +  e'}  =  R  (i  +  i'}  and 
since  there  is  to  be  no  reflection,  both  e'  and  i'  must  be  zero; 
whence 


Although  the  cases  considered  have  been  chosen  merely  to 
explain  general  principles,  and  do  not  exactly  represent  conditions 
likely  to  arise  in  practice,  they  should  nevertheless  be  helpful 
in  giving  some  indication  of  when  troublesome  surges  or  oscilla- 
tions are  likely  to  occur.  Instead  of  a  detached  "wave  pulse" 
travelling  along  a  line,  we  must  usually  think  of  a  "wave  train" 
of  harmonic  functions  of  gradually  decreasing  amplitude  travel- 
ling along  the  line  in  both  directions  from  the  point  where  the  dis- 
turbance occurs.  Arriving  at  the  ends  of  the  line,  or  at  points 
where  branch  circuits  or  transformers  are  connected,  these 
travelling  waves  of  energy  may  be  totally  or  partially  reflected. 
The  reflected  waves  meeting  the  outgoing  waves  may  lead  to 
considerable  magnification  of  the  original  trouble. 

1  Both  i'  and  e'  (the  "reflected"  current  and  voltage)  may  be  positive  or 
negative  quantities  depending  upon  their  direction  relatively  to  the  direction 
of  these  components  of  the  original  energy  wave  pulse. 


PROTECTION  AGAINST  LIGHTNING  173 

Standing  Waves. — If  the  conductor  resistance  is  so  small  as 
to  be  negligible,  the  oscillations  of  energy  may  be  thought  of  as  a 
wave  train  of  harmonic  functions  of  constant  amplitude  travelling 
to  the  end  of  the  line  which  we  shall  suppose  to  be  unloaded 
(i. e. ,  open) .  The  waves  are  therefore  reflected  in  the  manner  pre- 
viously explained  under  Case  (1),  and  the  reflected  waves, 
meeting  the  outgoing  waves,  produce  nodes,  or  points  of  zero 
potential,  and  antinodes  where  the  voltage  is  exactly  double  the 
maximum  voltage  of  the  original  disturbance.  At  these  points 
of  double  voltage  the  current  must  be  zero — since  the  total 
energy  remains  constant — and  the  nodes  of  the  current  waves 
therefore  occur  at  the  antinodes  of  the  voltage  waves.  The  com- 
bination of  the  outgoing  and  reflected  waves  is  thus  seen  to 
produce  standing  waves  which  remain  stationary  in  position 
although  varying  in  amplitude.  The  resistance  of  the  conductors 
prevents  this  simple  mode  of  oscillation  being  exactly  realized 
on  a  practical  transmission  line. 

87.  Line  Disturbances  Caused  by  Switching  Operations. — 
It  is  hardly  necessary  to  add  anything  to  what  has  already  been 
said,  in  order  to  emphasize  the  possible  danger  of  suddenly 
switching  a  source  of  electrical  energy  on  or  off  a  long  trans- 
mission line.  Unfortunately  the  calculations  of  the  probable 
surges  or  oscillations  are  not  easily  made,  and  moreover  accurate 
data  concerning  the  characteristics  of  the  various  circuits  and 
apparatus  connected  to  the  system  are  rarely  available.  It 
follows  that  the  engineer  cannot  predetermine  accurately  what 
will  happen  under  the  different  probable  or  possible  conditions 
of  operation;  but  a  general  understanding  of  the  principles 
underlying  the  creation  of  energy  surges  in  a  system  of  electric 
conductors  will  enable  him  to  avoid  obvious  mistakes  in  the 
design  and  operation  of  a  particular  transmission  scheme. 

There  is  frequently  danger  of  abnormally  high  voltages  due  to 
surges  at  the  points  where  there  is  a  change  in  the  constants  of  the 
circuit.  Thus,  if  a  transformer  is  connected  across  the  ends  of  a 
long  overhead  transmission  line,  there  will  be  a  rise  of  pressure 
when  a  travelling  wave  arrives  at  this  point  because  the  surge 


impedance  I .» /— )  of  the  transformer  winding  may  be  between 

2000  and  4000  ohms,  which  is  very  much  higher  than  that  of  the 
line  itself  (about  400  ohms  as  previously  explained).     For  this 


174  ELECTRIC  POWER  TRANSMISSION 

reason  the  end  turns  of  the  transformer  primaries  should  be 
specially  insulated  to  withstand  much  higher  voltages  between 
turns  than  the  remainder  of  the  winding. 

In  the  case  of  a  change  from  underground  to  overhead  trans- 
mission, a  surge  originating  in  the  cable  will  produce  a  rise  in 
pressure  at  the  junction  with  the  overhead  line,  while  the  con- 
trary will  occur  (i.e.,  the  voltage  will  be  reduced)  if  the 
surge  is  originated  in  the  overhead  line  and  passes  into  the  cable 
system  of  which  the  surge  impedance  will  always  be  smaller 
than  that  of  the  overhead  transmission. 

With  the  good  insulation  provided  on  modern  high  voltage 
systems,  it  is  doubtful  if  the  interruption  of  the  current  by 
opening  switches  under  load  is  likely  to  cause  serious  voltage 
disturbances,  except  in  the  case  of  air-break  switches  where  a 
long  arc  may  be  formed  and  suddenly  interrupted — as  for  in- 
stance by  a  draught  of  air — when  the  current  is  of  considerable 
value.  Oil-break  switches  almost  invariably  open  the  circuit 
at  the  instant  when  the  current  is  passing  through  zero  value. 

88.  Lightning. — The  foregoing  considerations  do  not  take 
into  account  the  effects  of  lightning,  either  by  direct  stroke  or 
by  induction,  because  in  such  cases  a  pressure  from  an  outside 
source  is  impressed  upon  the  circuit,  and  the  potential  of  these 
atmospheric  charges  may  be  tens  of  times  greater  than  any  surge 
voltage  due  to  a  redistribution  of  the  energy  stored  in  the  circuit 
itself. 

Although  our  knowledge  of  lightning  phenomena  is  still  far 
from  complete,  it  is  generally  agreed  that  a  single  stroke  of 
lightning  is  of  short  duration,  frequently  not  exceeding  the  one- 
thousandth  part  of  a  second.  If  an  overhead  conductor  re- 
ceives a  direct  stroke  of  lightning,  the  potential  value  of  the 
lightning  charge  is  generally  so  enormously  in  excess  of  the 
working  pressure  on  the  conductors  that  the  lightning  leaps 
over  the  insulators  down  the  pole  to  ground.  Any  charge  on 
the  line,  which  is  not  sufficiently  high  in  potential  above  ground 
to  jump  over  the  insulators,  will  travel  along  the  line  in  both 
directions  until  it  is  grounded  through  a  lightning  arrester  or  dis- 
sipated as  I2R  losses  in  the  conductors.  The  frequency  of  such 
travelling  waves  will  depend  upon  the  natural  frequency  of  the 
line,  and  may  be  of  the  order  of  1000  to  5000  cycles  per  second. 
If  the  resistance  of  an  arrester  or  the  path  through  which  a  dis- 
charge occurs,  were  zero,  the  current  passing  would  be  a  maxi- 


PROTECTION  AGAINST  LIGHTNING  175 

mum.  If  C  is  the  capacity  in  farads,  and  L  the  inductance, 
in  henrys,  of  unit  length  of  line,  then*/-  is  the  surge  impedance 
of  the  circuit;  and  the  maximum  possible  value  of  the  current 
will  be  7max.  =  E  -5-  -/-,  where  E  is  the  impressed  voltage,  which 

may  be  considered  as  something  less  than  the  pressure  which  will 
cause  a  flash-over  at  the  insulators. 

The  intense  concentration  of  lightning  disturbances  is  the 
cause  of  the  difficulties  experienced  in  protecting  transmission 
lines  by  means  of  lightning  arresters;  experience  tends  to  show 
that  an  arrester  does  not  adequately  protect  apparatus  at  a 
greater  distance  than  500  ft.,  yet  it  is  unusual  to  find  arresters 
on  a  transmission  line  at  closer  intervals  than  2000  ft. 

Disturbances  are  most  likely  to  occur  on  exposed  heights,  and 
on  open  wet  lowlands;  special  attention  should  therefore  be  paid 
to  lightning  protection  at  such  places. 

Although  the  quantity  of  electricty  in  a  lightning  flash  may  not 
be  very  great,  the  short  duration  of  the  flash  accounts  for  currents 
which  are  probably  of  the  order  of  20,000  to  50,000  amperes. 

Apart  from  the  effects  of  atmospheric  electricity,  it  is  necessary 
to  guard  against  the  abnormal  pressure  rises  that  will  occur  on 
long  transmission  lines  through  any  cause,  such  as  switching 
operations,  or  an  intermittent  "ground."  Over-voltages  up  to 
40  per  cent,  in  excess  of  the  normal  line  voltage  can  be  produced 
by  switching  in  a  long  line.  High-frequency  impulses  or  surges 
are  set  up,  which,  in  the  special  case  of  an  arcing  ground,  may 
give  rise  to  a  destructive  series  of  surges,  a  state  of  things  which 
will  continue  until  the  fault  is  removed.  An  arrester  which  may 
be  suitable  for  dealing  with  transitory  lightning  effects  may 
be  quite  inadequate  to  dissipate  the  charges  built  up  by  such 
continual  surges. 

89.  Protection  of  Overhead  Systems  against  Direct  Lightning 
Strokes  and  Sudden  Accumulations  of  High  Potential  Static 
Charges. — Under  this  heading  the  ordinary  lightning  rod  and 
grounded  guard  wire  will  be  briefly  dealt  with.  If  no  guard  wire 
is  used,  lightning  rods  should  be  provided  at  intervals  along  the 
line.  They  may  be  fixed  to  every  pole  or  tower,  but,  in  any  case, 
they  should  not  be  spaced  farther  apart  than  300  to  400  ft.  un- 
less the  spacing  of  the  supporting  poles  or  towers  has  to  be  greater 
than  this,  for  economic  reasons.  It  is  especially  important  to 


176  ELECTRIC  POWER  TRANSMISSION 

provide  them  on  the  poles  or  towers  in  exposed  positions  such  as 
hill  tops.  They  should  project  from  3  to  6  ft.  or  more  above 
the  topmost  wire.  A  convenient  form  of  lightning  rod  is  a 
length  of  galvanized  angle  iron  bolted  to  pole  top  or  forming 
an  extension  to  the  structure  of  a  steel  tower.  Long  lines  have 
been  worked  satisfactorily  for  extended  periods  without  lightning 
rods  or  guard  wires,  but  these  are  extra  high  pressure  transmis- 
sions which,  on  account  of  the  better  insulation  throughout,  are 
always  less  liable  to  trouble  from  lightning  than  the  lower 
voltage  systems. 

Although  engineers  are  still  divided  in  opinion  as  to  the  value 
of  the  protection  afforded  by  overhead  grounded  guard  wires, 
carried  the  whole  length  of  the  line  above  the  conductors,  it  is  now 
generally  recognized  that  this  method  of  protection  is  efficient. 
The  objections  to  the  guard  wire  are  the  additional  cost  and  the 
possibility  of  the  grounded  wire  breaking,  and  falling  across  the 
conductors  below,  thus  causing  an  interruption  to  continuous 
working.  Trouble  due  to  this  cause  is,  however,  exceedingly 
rare. 

It  has  been  suggested  that  the  guard  wire  or  wires  should 
be  of  the  same  material  as  the  conductors,  in  order  that  the  "life" 
of  all  the  wires  may  be  the  same;  but  there  are  other  considera- 
tions in  favor  of  using  a  galvanized  stranded  steel  cable  for  the 
guard  wire.  This  may  be  the  ordinary  cable,  ^{  Q  to  J{g  in-  in 
diameter,  as  used  for  guying  poles;  but,  where  great  strength  is 
required,  the  Siemens-Martin  steel  cable,  with  or  without  hemp 
core,  is  preferable.  Bessemer  steel  wire  has  not  been  found  satis- 
factory for  this  purpose.  In  the  case  of  the  "flexible"  steel 
tower  type  of  line,  a  strong  steel  guard  wire  joining  the  tops  of 
the  towers,  adds  greatly  to  the  strength  and  stability  of  the  line, 
and  may  even,  on  long  lines,  save  its  cost,  by  allowing  the  use  of 
lighter  structures  and  fewer  intermediate  (dead-ending)  towers. 

In  regard  to  the  position  of  overhead  guard  wires  relatively  to 
the  conductors,  it  is  obvious  that  a  number  of  grounded  wires 
surrounding  the  conductors  will  afford  better  protection  than  a 
single  wire  above  the  conductors;  and  two  guard  wires  are  some- 
times provided;  but  the  additional  cost  is  rarely  justified.  Per- 
fect protection  cannot  be  obtained  even  with  two  guard  wires, 
and  cases  have  been  reported  of  lightning  missing  the  grounded 
wire  and  striking  a  conductor  situated  immediately  below. 

The  best  position  for  a  single  guard  wire  placed  above  the  con- 


PROTECTION  AGAINST  LIGHTNING 


177 


ductors  is,  according  to  Dr.  Steinmetz,1  such  that  all  the  current 
carrying  wires  are  included  within  an  angle  of  60  degrees  below 
the  guard  wire.  Additional  wires  can  be  installed  in  exposed 
positions,  such  as  the  summit,  or  very  near  the  summit,  of  a  range 
of  hills,  or  by  the  shores  of  lakes  or  seas  where  the  prevailing 
winds  come  over  the  water.  In  such  positions,  an  additional 
guard  wire  on  the  side  of  the  conductors  may  be  useful.  The 
guard  wire  should  preferably  be  grounded  at  every  pole,  or  at 
least  every  500  ft. 

90.  Protection  of  Insulators  from  Power  Arcs. — As  a  special 
means  of  protecting  insulators  from  the  flash-over  caused  by 


FIG.  65. — Arcing  ring  on  pin  type  insulator. 

lightning,  or  the  power  arc  following  a  high  potential  discharge, 
the  "arcing  rings"  first  introduced  by  Mr.  L.  C.  Nicholson,  may 
be  mentioned.  These  rings,  which  are  grounded,  are  placed  in 
such  a  position  as  to  take  the  arc,  and  hold  it  at  a  sufficient  dis- 
tance from  the  porcelain  of  the  insulator  to  prevent  cracking  or 
breakage  by  heat.  The  illustrations,  Figs.  65  and  66,  show 
the  arrangement  of  the  grounded  arcing  rings  attached  to  stand- 
ard types  of  insulator  made  by  the  Locke  Insulator  Manufactur- 
ing Company.  It  is  not  claimed  that  these  rings  will  protect  an 

1  Discussion  of  the  Committee  on  "Lightning  Protection,"  of  the  National 
Electric  Light  Association,  May,  1908. 
12 


178 


ELECTRIC  POWER  TRANSMISSION 


insulator  against  a  direct  lighting  stroke;  but  their  utility  on 
high-pressure  lines  transmitting  large  amounts  of  power  has  been 
proved  without  doubt. 

Although  a  pair  of  metal  rings,  one  near  the  top  and  one  near 
the  bottom  of  the  insulator  will  probably  afford  the  best  pro- 
tection, the  arrangement  of  arcing  rods  shown  in  Fig.  67  will 
prove  almost  as  effective;  the  object  being  not  only  to  provide  a 
path  for  the  high-voltage  flash-over  which  shall  keep  the  arc 
away  from  the  porcelain,  but  also  to  prevent  puncture  of  the 
insulator  due  to  concentration  of  potential  at  the  points  of 


,  Arcing  Ring  No.2838 


Ring  No.2839 


Stanislaus  Clamp 


Plan  of  Ring  No.2839 
FIG.  66. — Arcing  rings  on  suspension  type  insulator. 

attachment  to  the  metal  fittings.  The  arcing  horns  as  shown 
in  Fig.  67  are  used  on  the  insulators  of  the  66,000  volt  lines  of 
the  Peninsular  Power  Gompany,  of  Iron  Mountain,  Mich.,  de- 
scribed by  Mr.  Max.  H.  Collbohm  in  the  Electrical  World  of  April 
18,  1914. 

91.  Methods  of  Grounding. — The  ground  wire  from  lightning 
rod,  guard  wire,  or  arrestor,  on  high-tension  transmission  circuits, 
should  be  as  short  and  as  straight  as  possible;  it  need  not  be  of 
very  low  resistance;1  but  small  reactance  is  of  first  importance. 

1  The  question  of  resistances  in  series  with  spark  gap  arresters  will  be 
taken  up  in  Article  93. 


PROTECTION  AGAINST  LIGHTNING 


179 


The  ground  plate  should  have  a  large  surface  but  the  material  is 
of  little  importance,  except  that  it  is  not  wise  to  bury  aluminum 
wires  in  the  ground,  because  of  possible  electrolytic  action. 
Galvanized  iron  is  a  good  material.  If  the  ground  contact  is 
made  with  one  or  more  iron  pipes  buried  or  driven  into  the 
ground,  these  pipes  may  be  from  1  to  1^  in.  in  diameter,  and 
a  good  connection  should  be  made  to  the  top  of  the  pipe,  as  the 
inductive  effect  of  an  iron  tube  surrounding  the  ground  wire 
might  be  considerable  if  a  connection  were  made  only  at  the 
bottom  of  the  pipe.  One  or  more 
pipes  8  to  10  ft.  long,  driven  into 
the  ground  with  6-in.  to  12-in.  pro- 
jecting above,  will  generally  be 
found  more  effective  than  buried 
plates.  A  very  low  resistance 
ground  is  not  essential  on  a  high- 
tension  system,  and,  generally 
speaking,  the  special  forms  of 
ground  plate  made  of  perforated 
copper,  designed  to  hold,  or  to  be 
in  contact  with,  crushed  charcoal, 
are  unnecessary.  If  a  plate  is  used, 
this  should  be  not  less  than  12  in. 
square,  but  need  not  be  larger  than 
18  in.  square;  it  maybe  of  galva- 
nized iron  <K6  in.  Or  VA  in.  thick,  FIG.  67.— Arrangement  of  arcing 
.  .  ,  .,  ,  .  ,,  rods  above  and  below  string  of 

buried   as  deep  as  possible  in  the    insuiators. 

ground,  and,  in  all  cases,  an  effort 

should   be   made  to  secure  permanently  damp  soil  for  ground 

plates  or  pipes. 

92.  Relieving  Conductors  of  High  Potential  "Static." 
Water  Jet  Arresters.— By  directing  a  stream  of  water  from 
the  nozzle  of  a  grounded  metal  pipe  on  to  the  high-tension  con- 
ductors, a  high-resistance  non-inductive  path  to  ground  is  pro- 
vided for  the  extra-high  potential  charges  on  the  line;  but  there 
will  be  very  little  leakage  of  power  current.  It  is  claimed  that 
arresters  constructed  on  this  principle  have  been  found  useful  in 
practice;  but  the  employment  of  jets  of  water  has  its  objections. 
It  is  usual  to  put  the  jets  in  action  only  at  times  when  electric 
storms  are  pending;  and  the  reliance  on  the  " human  element" 
renders  the  apparatus  less  valuable  than  an  equally  effective 


180  ELECTRIC  POWER  TRANSMISSION 

device  which  is  always  ready  to  act.  Patents  have  been  granted 
for  various  forms  of  water-jet  arresters,  but  they  are  not  exten- 
sively used  at  the  present  time.  The  chief  function  of  the  water 
jet  is  to  prevent  the  building  up  of  static  pressures  on  the  line 
caused  by  the  contact  of  dust,  snow,  or  rain  drops,  blown  against 
or  falling  upon  an  insulated  line  of  considerable  length,  or  by 
variations  in  the  potential  of  the  atmosphere,  at  different  parts 
of  a  line  traversing  hilly  country. 

Since  other  means,  such  as  highly  inductive  resistances,  or 
the  grounding  of  the  neutral  point  of  transformers  connected 
to  the  line,  are  available  for  preventing  the  accumulation  of  static 
charges  on  an  overhead  line,  the  advantages  claimed  for  water 
jets  are  not  obvious.  Many  engineers  of  high  standing  in  Europe 
use  water-jet  dischargers,  and  recommend  them  for  pressures  up  to 
40,000  and  even  50,000  volts.  For  higher  line  pressures,  iron- 
cored  choke  coils  are  preferable  for  the  purpose  of  draining  the 
system  of  static  charges.  These  should  always  be  of  the  single- 
phase  type  connected  between  the  line  and  ground  through 
damping  resistances,  the  purpose  of  which  is  to  prevent  energy 
oscillations.  The  connection  to  the  line  should  be  made  at  a 
point  between  the  lightning  arrester  and  the  apparatus  to  be 
protected.  It  has  been  stated  that  water  jets  are  effective  in 
preventing  damage  by  direct  lightning  strokes  in  their  immedi- 
ate neighborhood;  but  no  very  definite  evidence  to  this  effect 
is  forthcoming.  If  the  column  of  water  is  of  so  high  a  resistance 
that  it  causes  no  appreciable  leakage  of  the  power  current,  it  is  not 
easy  to  believe  that  it  can  prove  equal  in  an  emergency  to  shunt-^ 
ing  the  hundreds  or  thousands  of  amperes  which  would  otherwise 
pass  to  ground  through  the  apparatus  connected  to  the  line. 

93.  Horn  Gap. — Nearly  all  lightning  arresters  are  designed  on 
the  principle  of  one  or  more  spark  gaps  between  the  conductors 
and  ground,  the  air  space  being  so  adjusted  that  the  normal  differ- 
ence of  potential  between  the  line  and  ground  is  insufficient  to 
jump  the  gap;  but  abnormally  high  pressures  will  break  down  the 
insulation  of  the  gap,  and  so  find  a  path  to  ground  before  the 
pressure  is  sufficiently  high  to  damage  the  insulation  of  the  line 
or  the  apparatus  connected  thereto. 

The  ordinary  horn  gap  arrester  of  the  type  shown  in  Fig.  68, 
is  so  well  known  that  it  requires  no  detailed  description.  When 
the  potential  rises  to  such  a  value  that  it  can  jump  the  gap 
at  the  base  of  the  curved  wires,  the  power  arc  will  follow  the  dis- 


PROTECTION  AGAINST  LIGHTNING 


181 


charge,  but,  owing  partly  to  the  upward  tendency  of  the  heated 
air,  and  mainly  to  the  magnetic  field  produced  by  the  current 
itself,  the  arc  is  driven  upward  toward  the  ends  of  the  "horns" 
where,  after  being  sufficiently  drawn  out  in  length,  it  is  finally 
ruptured.  The  horn  gap  is  not  effective  when  set  to  discharge 
at  pressures  below  13,000  volts,  because,  with  a  small  gap  (less 
than  1  in.),  the  arc  may  not  rise  and  break  properly.  The  usual 
settings  for  horn  gaps  are  as  follows :  the  voltages  in  the  table  are 
the  r.m.s.  values  on  the  sine-wave  assumption,  and  they  must  not 


FIG.  68. — Horn  arrester  with  disconnecting  switch. 

be  used  except  as  a  rough  indication  of  the  probable  gap  between 
horns : 


Gap,  in. 

Working  voltage  across  gap 

Spark-over  voltage, 
approximate 

1 

21,000 

30,000 

IK 

29,000 

41,000 

2 

34,000 

48,000 

3 

44,000 

63,000 

4 

52,000 

74,000 

6 

65,000 

93,000 

8 

77,000 

110,000 

10 

88,000 

126,000 

12 

98,000 

140,000 

A  non-inductive  resistance  should  be  connected  in  the  ground 
wire  from  the  horn  arrester.  An  ordinary  wooden  barrel  filled 
with  water,  with  a  connecting  plate  at  the  bottom,  and  the  upper 
terminal  carried  about  6  in.  below  the  surface  of  the  water, 
makes  an  effective  resistance.  If  no  resistance  is  provided  in  the 
ground  connection,  the  momentary  discharge  of  the  power 


182  ELECTRIC  POWER  TRANSMISSION 

current  may  be  excessive,  dangerous  surges  may  be  set  up  in  the 
line,  and  there  is  the  possibility  of  synchronous  machines  being 
thrown  out  of  step. 

As  an  example  of  how  a  suitable  value  for  the  resistance  in 
the  ground  connecton  may  be  estimated,  consider  a  60,000  volt 

three-phase   line  of  surge  impedance   -ft  =  300  ohms.     The 

maximum  value  of  the  working  pressure  between  wire  and  ground 
under  normal  conditions,  on  the  sine  wave  assumption,  is 

60000 

#n(max.)  =  -— ^-  X  V2  volts. 

Let  the  safety  factor  of  the  line  insulators  be  3,  that  is  to  say, 

60000 
the  insulators  will  flash  over  with  — -/=-  X    3  volts  (r.m.s.)  to 

ground.  Any  voltage  exceeding  this  value  will  cause  a  "spill- 
over" and  it  therefore  follows  that  the  maximum  possible  tran- 
sient voltage  which  can  be  added  to  the  crest  of  the  normal 
e.m.f.  wave  and  travel  along  the  line,  is 

/60000          ^\ 
e  =  2l — 7=-  X  \/2  )  =  98000  volts. 

The  maximum  transient  current  that  the  arrester  will  have  to 
take  care  of  is  therefore 

98000 
"•*  =  "300"  =  327  amperes' 


If  the  arrester  is  to  prevent  a  spill-over  and  afford  reasonable 
protection  to  the  insulation  of  the  station  apparatus,  it  must  be 
set  for  a  lower  voltage  than  3En,  the  usual  setting  being  about 
1.5En.  If  Ra  is  the  non-inductive  resistance  in  the  grounded 
connection  of  the  arrester,  the  potential  drop  across  it,  if  the 
maximum  possible  value  of  the  surge  current  passes  through 
it  to  ground,  will  be  iRa,  whence 

1.5  En  =  iRa  +  En 

„        0.5En       17000 
and  .Ra  =  — : —  =  — ,==-  =  52  ohms. 

1  o£i 

In  this  calculation,  the  resistance  of  the  air-gap  is  supposed  to 
be  zero  at  the  time  of  the  discharge;  but  in  any  case  in  does 
not  follow  that  52  ohms  will  be  the  best  possible  value  for  the 


PROTECTION  AGAINST  LIGHTNING  183 

resistance:  the  problem  is  not  quite  so  simple  as  these  considera- 
tions would  appear  to  imply.  Experience  is  the  best  guide  in 
arriving  at  the  most  effective  value  of  the  resistance  in  the 
ground  connection.  It  is  evident  that  an  appreciable  amount  of 
resistance  is  desirable,  but  that  too  high  a  resistance  will  allow 
insufficient  current  to  pass  to  ground,  and  moreover  may  stand 
in  the  way  of  the  rapid  extinction  of  the  arc  across  the  horns. 
When  deciding  upon  a  suitable  value,  the  effect  of  the  power 
current  following  the  discharge  should  be  considered.  Thus, 
in  the  above  example  the  power  current  on  one  phase  (conductor 

to  ground)  would  be  -7=—  r^  =  666  amperes,  representing  an 
V  3  X  52 

output  of  23,000  kw.  on  one  phase,  if  the  arrester  is  near  the 
generating  station.  For  this  reason  alone,  it  might  be  desirable 
to  use  a  grounding  resistance  somewhat  larger  than  the  calculated 
52  ohms. 

One  serious  disadvantage  to  the  ordinary  horn-gap  arrester 
is  the  liability  of  an  intermittent  arc  setting  up  surges  and  high 
potential  disturbances  which  may  lead  to  more  trouble  than  the 
original  cause  of  the  spark-over.  Fairly  satisfactory  results 
have  been  obtained  by  providing  a  number  of  horn  gaps  on  a 
high-tension  transmission  and  "grading"  these,  by  adjusting 
some  of  them  to  discharge  with  a  very  small  rise  of  pressure 
through  a  high  resistance;  while  other  sets  would  have  larger 
gaps  and  lower  resistances  in  series;  the  very  largest  gap  being 
such  as  to  break  down  only  rarely,  under  exceptionally  high  pres- 
sures, and  this  should  have  a  very  low  resistance  in  series  but 
may  with  advantage  be  protected  by  a  fuse. 

Horn  arresters,  if  intelligently  placed  and  properly  connected 
and  adjusted,  are  capable  of  affording  good  protection  on  circuits 
(both  A.C.  and  D.C.)  up  to  20,000  volts;  but  the  multi-gap  and 
"low  equivalent"  arresters,  as  originally  suggested  by  Mr. 
P.  H.  Thomas,  have  some  special  features  which  have  led  to  their 
frequent  adoption  on  alternating  current  circuits  for  pressures 
up  to  about  35,000  volts. 

94.  Multiple-gap  Low  Equivalent  Arrester. — In  this  type  of 
arrester  there  are  many  air  gaps  in  series  between  the  line  and 
ground.  No  single  gap  is  greater  than  ^2  m-  or  He  m-  and 
it  occurs  between  the  adjacent  surfaces  of  small  cylinders  made 
of  a  so-called  "non-arcing"  metal  as  used  in  the  earlier  types 
of  Wurtz  arrester.  The  number  of  gaps  in  series  depends  upon 


184 


ELECTRIC  POWER  TRANSMISSION 


the  working  voltage  of  the  line,  and  the  last  of  the  metal  cylinders 
is  connected  to  ground  (or  to  one  of  the  return  conductors,  as 
the  case  may  be)  though  a  non-inductive  resistance,  which  may, 
with  advantage,  be  shunted  by  a  fuse  in  series  with  a  spark  gap. 
Sometimes  a  portion  of  this  resistance  is  bridged  by  a  number  of 


Ground  Shield 
Recommended  for  Arresters 
on  the  higher  Voltage  Systems 


Spark  Gap  and  Fuse 
(Not  Essential) 


Series  Resistance 


FIQ.  69. — Diagram  of  multiple-gap  low  equivalent  arrester. 


spark  gaps,  all  as  shown  in  the  diagram  Fig.  69.  These  shunted 
gaps  act  as  a  sort  of  by-pass  for  heavy  discharges,  the  amount 
of  the  series  resistance,  through  which  all  discharges  have  to 
pass,  being  comparatively  small.  The  theory  of  the  low  equiva- 


PROTECTION  AGAINST  LIGHTNING  185 

lent  arrester  has  been  ably  discussed  by  other  writers.1  Its 
action  is,  briefly,  as  follows.  There  is  a  certain  electrostatic 
capacity  between  consecutive  cylinders,  and  between  each  one 
of  these  cylinders  and  ground;  and  the  potential  gradient  is  con- 
siderably greater  at  the  high  voltage  end  of  the  arrester,  with  the 
result  that,  when  the  total  voltage  across  the  arrester  reaches  a 
certain  critical  value,  the  breakdown  occurs  between  the  first 
and  second  cylinders.  The  second  cylinder  is  then  connected 
to  the  first  by  an  arc,  so  that  its  potential  rises  accordingly, 
until  a  breakdown  occurs  between  the  second  and  third  cylinders; 
and  so  on.  The  line  current  then  follows  the  discharge,  and  in  so 
doing,  tends  to  produce  a  uniform  fall  of  potential  along  the  line 
of  cylinders,  with  the  result  that  the  maximum  potential  difference 
between  cylinders  is  considerably  less  than  that  required  for 
the  initial  breakdown,  and  the  power  arc  is  ruptured  as  the 
current  passes  through  zero  value.  When  a  breakdown  occurs 
between  two  cylinders,  the  potential  of  the  lower  cylinder  of 
the  series  will  depend  upon  the  quantity  of  electricity  which 
passes  to  it  from  the  more  highly  charged  cylinder.  The  initial 
current  is  really  a  capacity  current,  and  it  will  therefore  be 
greater  at  the  higher  frequencies;  but,  by  ^  scientific  propor- 
tioning of  the  shunted  resistance,  a  very  satisfactory  arrester 
of  this  type  can  be  made,  for  use  on  circuits  up  to  about  13,000 
volts;  it  is  less  effective  on  higher  voltages,  but  is  actually  used 
on  20,000-volt,  and  even  35,000-volt  transmission  lines. 

One  reason  why  the  multiple-gap  low  equivalent  arrester  is 
not  satisfactory  on  very  high  voltage  systems  is  that  the  necessary 
increase  in  the  number  of  gaps  to  prevent  arcing  over  by  the  line 
voltage  alone,  is  out  of  all  proportion  to  the  increase  in  voltage. 
There  is  also  much  uncertainty  as  to  the  number  of  gaps  required, 
which  will  depend  on  the  position  of  the  arrester  relatively 
to  surrounding  grounded  objects.  With  the  ground  potential 
brought  very  near  to  the  arrester,  the  potential  gradient  at  the 
end  near  the  line  frequently  becomes  high  enough  to  ionize  the 
air  between  the  cylinders,  thus  carrying  the  line  potential  to 
lower  cylinders,  until  the  remaining  gaps  are  so  few  that  a  dis- 
charge occurs.  In  order  to  obtain  the  more  equal  division  of 

1  See  Dr.  Steinmetz  on  the  theory  of  this  type  of  arrester  in  vol.  xxv 
(1906),  of  the  Proceedings  of  the  A.  I.  E.  E.  Also  the  excellent  book  "Les 
sur tensions  dans  les  distributions  d'energie  electrique,"  by  I.  Van  Dam 
(Van  Mantgem  &  De  Does,  Amsterdam)  whose  theoretical  treatment  of 
this  and  kindred  problems  always  has  the  practical  end  in  view. 


186  ELECTRIC  POWER  TRANSMISSION 

the  total  potential  difference,  and  so  allow  of  a  reduction  in  the 
total  number  of  gaps,  such  as  would  be  obtained  by  removing 
the  whole  arrester  to  a  considerable  distance  from  grounded 
objects,  a  metal  guard  plate  or  shield  is  sometimes  placed  near 
the  gaps  at  the  high  potential  end  of  the  arrester,  and  connected 
to  the  line  wire  as  indicated  in  Fig.  69.  The  theory  of  the  poten- 
tial distribution  over  the  string  of  insulated  metal  cylinders 
in  the  multiple-gap  arrester  need  not  be  discussed  here  because 
the  problem  has  already  been  considered  in  connection  with  the 
suspension  type  of  insulator  (refer  Article  74,  Chapter  V).  It 
is  evident  that,  if  the  electrostatic  capacity  between  the  con- 
secutive elements  of  the  arrester  can  be  made  large  relatively 
to  the  capacity  to  ground,  a  more  uniform  drop  of  potential 
over  the  series  of  elements  will  result,  thus  rendering  this  type  of 
arrester  more  suitable  for  the  higher  pressures.  The  introduction 
of  condensers  to  increase  the  capacity  from  element  to  element 
is  the  basis  of  the  Moligniani  system  which  has  been  patented 
and  is  in  use  in  Italy.  It  seems  probable,  however,  that  the 
high  cost  of  the  condensers  will  stand  in  the  way  of  this  method 
proving  superior  to  other  possible  alternatives. 

95.  Spark-gap  Arresters  with  Circuit  Breakers  or  Re-setting 
Fuses. — If  the  resistance  in  series  with  a  gap  arrester  is  very 
small  a  good  path  is  provided  to  ground  for  taking  a  very  heavy 
discharge;  but  there  will  be  a  large  flow  of  power  current  in  the 
arc  following  the  discharge.  This  current  may  be  interrupted 
by  connecting  some  self-acting  device  such  as  a  fuse  or  automatic 
circuit  breaker  in  the  ground  connection;  and  arresters,  whether 
of  the  horn  type  or  with  any  other  kind  of  spark  gap,  are  some- 
times provided  with  fuses  so  arranged  that  when  one  fuse  blows; 
the  dropping  of  a  lever  or  an  equivalent  device,  automatically 
inserts  another  fuse,  so  that  the  system  is  not  left  unprotected. 
Even  without  automatic  replacement,  if  a  number  of  gaps  with 
fuses  are  connected  in  parallel,  it  will  generally  be  found  that  one 
discharge  will  not  blow  all  the  fuses,  and  that  during  the  passage 
of  a  single  storm,  the  line  will  be  adequately  protected. 

In  the  Garton-Daniels  arrester,  for  use  on  alternating-current 
circuits  up  to  20,000  volts,  the  principle  of  the  multiple  gap  is 
combined  with  a  simple  type  of  automatic  circuit  breaker  con- 
nected as  a  shunt  to  some  of  the  spark  gaps,  in  order  that  the 
discharge  path  for  the  lightning  shall  remain  unaltered  even 
during  the  operation  of  the  arrester.  The  arrester  is  built  up 


PROTECTION  AGAINST  LIGHTNING 


187 


of  several  unit  parts  connected  in  series;  each  unit  being  rated 
for  3300  volts.  The  illustration,  Fig.  70,  shows  a  complete 
single-phase  arrester  for  10,000  volts.  On  a  20,000  volt  circuit, 
there  would  be  eight  units  in  series,  the  total  air  gap  distance 
being  l^f  in.,  with  a  series  resistance  averaging  3800  ohms. 
The  diagram,  Fig.  71,  refers  to  a  single  unit  of  the  Garton-Daniels 
arrester.  The  discharge  follows  the  straight  path  through  the 
two  sets  of  air  gaps  and  the  resistance  rod,  as  indicated  by  the 
round  dots.  The  power  current  following  the  discharge  will, 
after  passing  through  the  two  upper  gaps  and  the  resistance  rod, 
be  shunted  by  the  low  resistance  winding  of  the  circuit  breaker; 


FIG.  70. — Garton-Daniels  arrester 
for  10,000-volt  circuit. 


FIG.  71. — Diagram  .of  Garton-Daniels 
arrester. 


and  if  this  following  current  is  too  heavy  to  be  ruptured  by  the 
combined  action  of  these  two  gaps  and  the  resistance  rod,  the 
iron  armature  of  the  circuit  breaker  will  be  lifted  by  the  action 
of  the  solenoid,  thus  throwing  the  two  lower  spark  gaps  in  series, 
and  extinguishing  the  arc. 

96.  Aluminum  Cell  Arrester. — When  two  aluminum  electrodes 
are  immersed  in  a  suitable  electrolyte,  an  insulating  film  of  hy- 
droxide of  aluminum  is  formed  on  the  surface  of  the  metal; 
this  effectually  prevents  the  passage  of  any  appreciable  amount  of 


188  ELECTRIC  POWER  TRANSMISSION 

current  until  a  certain  critical  voltage  is  reached,  when  the  film 
breaks  down  and  the  current  is  limited  only  by  the  resistance 
of  the  electrolyte.  On  lowering  the  voltage,  the  film  is  reformed 
and  the  flow  of  current  again  limited  to  a  very  small  amount. 

With  alternating  currents,  the  critical  potential  difference  per 
pair  of  plates  is  about  350  volts,  and  the  practical  construction  of 
lightning  arresters  on  this  principle  consists  in  stacking  a  large 
number  of  cone-shaped  aluminum  plates  one  within  the  other, 
with  suitable  separating  washers  of  insulating  material  between 
them.  In  this  manner  a  column  is  formed  of  a  large  number  of 
cells  in  series,  capable  of  withstanding  high  voltages.  The  whole 
is  enclosed  in  a  case  containing  oil,  which  improves  the  insulation 
and  prevents  the  evaporation  of  the  electrolyte  which  fills  the 
spaces  between  adjacent  trays  within  a  short  distance  of  the  edge. 

If  cells  built  up  in  this  manner  are  connected  directly  between 
line  and  ground,  there  will  be  an  appreciable  current  passing 
through  them,  which  is  partly  a  leakage  current,  but  chiefly  a 
capacity  current.  It  is  therefore  customary  to  insert  a  spark 
gap,  usually  of  the  horn  type,  in  series  with  the  aluminum  cell 
arrester;  the  gap  being  set  to  break  down  with  a  pressure  slightly 
in  excess  of  the  normal  working  voltage. 

Although  the  film  of  hydroxide  is  formed  on  the  plates  at  the 
factory  before  the  arresters  are  installed,,  it  is  necessary  to  main- 
tain it  by  periodic  "charging"  of  the  cells;  this  being  done  by 
closing,  or  nearly  closing,  the  spark  gap  in  series,  so  as  to  put  the 
full  line  pressure  across  the  arrester.  It  is  generally  recom- 
mended that  this  be  done  once  every  day. 

In  principle  the  aluminum  cell  arrester  would  appear  to  offer 
an  ideal  solution  of  the  problem  of  lightning  protection;  because, 
once  the  critical  voltage  is  exceeded,  and  the  film  broken  down,  a 
very  large  current — depending  on  the  amount  of  separation  and 
the  area  of  the  plates,  and  also  the  nature  of  the  electrolyte — is 
allowed  to  pass  to  ground;  and  the  device  is  capable  of  dealing 
with  continual  surges,  such  as  will  occur  with  an  intermittent 
ground,  for  a  period  of  about  half  an  hour  without  excessive  heat- 
ing. In  practice  it  has  proved  fairly  satisfactory,  especially  on 
the  higher  voltages;  but,  apart  from  its  large  initial  cost,  it  has  fre- 
quently been  found  to  be  somewhat  costly  in  upkeep,  as  the 
aluminum  cells  are  liable  to  become  damaged  through  frequent 
and  heavy  discharges,  and  have  to  be  periodically  reformed  or 
replaced.  Then  again,  the  necessity  of  charging  with  the  line 


PROTECTION  AGAINST  LIGHTNING  189 

current  is  an  objection  where  there  is  not  an  operator  constantly 
in  attendance;  and  lastly,  it  must  not  be  overlooked  that  the 
device  suffers  from  the  disadvantage  common  to  all  spark-gap 
devices,  namely  that  high-frequency  surges  are  liable  to  be  set  up 
in  the  system  when  the  spark  gap  discharges.  In  this  particular 
case  the  trouble  is  liable  to  occur,  not  only  when  the  horn  gap 
breaks  down  while  fulfilling  its  function  of  discharging  an  excess 
of  pressure  through  the  cells,  but  also  when  the  spark  is  delib- 
erately formed  for  the  purpose  of  charging  the  cells.  On  very 
high  pressure  systems  it  is  possible  that  the  surges  set  up  by 
spark  gaps  in  series  with  the  resistance  of  the  cells  are  not  likely 
to  cause  trouble;  but  this  suggests  the  possibility  of  simpler 
and  less  costly  devices,  such  as  the  graded  horn  gaps  previously 
referred  to,  being  equally  effective.  On  the  other  hand,  when 
used  on  low  voltage  systems  operating  at  about  11,000  volts 
(especially  if  the  generators  are  directly  connected  to  the  trans- 
mission line,  without  the  intervention  of  step-up  transformers) 
the  operation  of  charging  the  aluminum  cell  arresters  in  the 
generating  station  has  been  known  to  break  down  the  insulation 
of  the  generators. 

The  makers  of  this  type  of  arrester  appear  to  have  recognized 
the  danger  of  damage  to  apparatus  arising  from  the  operation 
of  charging  the  cells,  and  they  now  recommend  that  the  charg- 
ing current  be  passed  through  a  resistance  in  series  with  the 
arresters.  Suitable  resistances  are  provided  in  connection  with 
the  horn  gaps  so  that  the  line  pressure  is  not  put  directly  across 
the  aluminum  arresters  at  the  time  of  charging. 

The  latest  development  in  this  direction  consists  of  a  some- 
what complex  arrangement  combining  sphere  gaps,  horn  gaps, 
and  resistances;  the  object  being  to  obtain  a  quick  and  free  dis- 
charge of  disturbances  of  steep  wave  front  across  sphere  gaps 
without  resistance  in  series.  If  the  arc  persists  across  this  gap  it 
rises  to  another  sphere  gap  which  has  resistance  in  series  to 
dampen  the  oscillations  of  energy.  Other  lightning  discharges  are 
dealt  with  by  a  horn  gap  of  the  ordinary  type  mounted  imme- 
diately above  the  second  sphere  gap. 

97.  Condensers. — Although  much  has  been  done,  and  more 
good  work  will  probably  be  done  in  the  future  by  the  intelligent 
"grading"  of  a  number  of  spark  gaps,  to  afford  a  path  to  ground 
and  yet  avoid  the  setting  up  of  dangerous  high-frequency  surges, 
the  objections  to  all  spark  gaps  are  (1)  the  necessity  of  an  appre- 


190  ELECTRIC  POWER  TRANSMISSION 

ciable  increase  in  pressure  above  normal  line  pressure  to  break 
down  the  resistance  of  the  gap,  and  (2)  the  danger  of  high-fre- 
quency oscillations  being  set  up  in  the  network  of  conductors. 

A  non-inductive  low-resistance  direct  connection  to  ground  can 
obviously  not  be  made  on  a  high-tension  alternating-current  over- 
head transmission  line;  but  a  path  to  ground  may  be  provided 
either  through  a  highly  inductive  choke  coil,  or  through  a  con- 
denser, or  both,  without  the  necessity  of  providing  a  spark  gap 
in  series.  The  inductive  resistance  may  easily  be  designed  to 
pass  only  an  inappreciable  current  of  normal  or  higher  frequency, 
and  it  will  therefore  be  useless  for  affording  relief  in  the  case  of 
high-frequency  surges;  but  it  is  capable  of  relieving  the  line  of 
slowly  accumulated  static  charges.  The  condenser,  however, 
acts  as  an  almost  perfect  insulator  so  far  as  direct  currents  are 
concerned;  but  it  is  pervious  to  high-frequency  currents,  and  a 
suitably  designed  condenser,  or  rather  battery  of  condensers, 
connected  between  line  and  ground  without  the  intervention 
of  any  spark  gap,  is  certainly  an  ideal  device  for  dealing  with 
the  very  high-frequency  oscillations  that  accompany  lightning 
phenomena.  This  is  the  chief  function  of  the  Mosciki  condensers 
which,  although  not  largely  used  on  this  continent,  have  found 
favor  in  Europe — where  they  have  been  in  use  for  many  years 
in  almost  every  country — and  also  in  South  Africa  and  China. 

The  travelling  waves  induced  by  a  lightning  discharge  on  a 
transmission  line  may  have  a  frequency  of  the  order  of  1000 
to  5000  cycles  per  second,  and  a  set  of  condensers  which  will 
pass  but  a  very  small  current  at  25  to  60  cycles,  will  deal  with 
much  larger  currents  on  these  higher  frequencies.  As  a  matter 
of  fact,  an  electric  discharge  between  cloud  and  ground  or 
between  cloud  and  cloud  may  induce  in  the  line  travelling  waves 
having  frequencies  considerably  in  excess  of  100,000  cycles  per 
second.  This  is  proved  by  the  fact  that  wireless  telegraphic 
apparatus  which  responds  only  to  frequencies  ranging  between 
about  100,000  and  1,000,000  cycles  per  second  is  interfered  with 
by  atmospheric  electric  storms.  A  40-amp.  fuse  in  series  with  a 
condenser  has  been  blown  during  atmospheric  discharges,  al- 
though the  condenser  could  not  possibly  pass  more  than  a 
hundredth  part  of  this  current  on  frequencies  of  3000  or  4000. 

It  is  perhaps  not  generally  understood  that  high-frequency 
travelling  waves  may  break  down  the  insulation  of  generators  or 
transformers  even  when  the  voltage  of  the  induced  charges  is 


PROTECTION  AGAINST  LIGHTNING  191 

small  as  compared  with  the  normal  operating  voltage.  The 
trouble  is  that  the  wave  is  short,  and  the  point  of  zero  potential 
may  be  only  a  few  hundred  feet  behind  the  point  of  maximum 
potential.  If,  therefore,  a  travelling  wave  of  this  nature  enters 
a  piece  of  electrical  machinery  such  as  a  generator  or  trans- 
former, the  full  difference  of  potential,  which  may  amount  to 
only  a  few  thousand  volts,  may  be  applied  across  adjacent  layers 
of  the  coil  winding,  thus  causing  a  puncture  and  ultimate  break- 
down of  the  insulation,  even  if  the  apparatus  as  a  whole  is 
insulated  to  withstand  pressures  of  100,000  to  200,000  volts  to 
ground.  As  a  protection  against  trouble  of  this  sort  from  high- 
frequency  induced  charges,  the  condenser  appears  to  offer  a  good 
solution. 

It  must  not  be  understood  from  these  notes  on  the  uses  of  con- 
densers as  lightning  arresters  that  the  discharge  is  diverted  to 
ground  through  the  condenser  and  so  dissipated,  much  as  energy 
would  be  dissipated  in  a  resistance;  because  the  condenser  can- 
not absorb  or  dissipate  any  but  the  smallest  percentage  of  the 
energy  passing  through  it.  The  energy  is  necessarily  re-delivered 
to  the  line  from  which  it  originally  came,  and  is  ultimately  dis- 
sipated through  the  ohmic  resistance  of  the  conductors.  The 
function  of  the  condenser  is,  in  fact,  somewhat  analogous  to  that 
of  an  air  chamber  on  a  water  pipe  in  which  the  rate  of  flow  is 
subject  to  sudden  variations. 

In  view  of  the  fact  that  the  condenser  merely  shunts  the  high- 
frequency  oscillations,  and  so  prevents  damage  to  the  apparatus 
to  be  protected,  but  returns  nearly  all  this  energy  to  the  line  where 
it  ultimately  dies  out  owing  to  the  conductor  resistance,  it  would 
seem  advisable  to  provide  some  small  amount  of  resistance  in 
series  with  the  condensers,  even  at  the  risk  of  slightly  higher 
surge  pressures  across  the  apparatus  to  be  protected.  The 
intelligent  combination  of  condensers,  reactance  coils,  and  resist- 
ances, may  be  expected  to  afford  good  protection;  but  there  is 
always  a  danger  of  resonance  effects  at  certain  critical  wave 
frequencies. 

Since  a  condenser  of  appreciable  capacity  is  obtainable  from  a 
comparatively  short  length  of  insulated  underground  cable,  it  is 
certainly  advantageous  to  lead  the  current  from  an  overhead 
transmission  into  generating-  and  sub-stations  through  a  length 
of  (say)  200  to  300  feet  of  underground  cable.  Of  course  this 
may  not  be  possible  in  the  case  of  very  high  pressures  because 


192  ELECTRIC  POWER  TRANSMISSION 

of  the  high  cost  of  the  cable,  but  as  a  matter  of  fact  any  form  of 
condenser  becomes  very  costly  when  designed  for  use  on  high 
voltage  systems.  The  effect  of  an  underground  cable  on  a  surge 
travelling  along  an  overhead  line  has  already  been  referred  to 
in  Article  87.  The  greater  capacity  (per  unit  length)  of  the 


cable  is  such  that  its  surge  impedance  *^  would  be  of  the  order 

of  40  ohms,  compared  with  about  400  ohms  for  the  overhead  line. 
98.  Spacing  of  Lightning  Arresters.  —  A  reasonable  distance 
must  be  allowed  between  the  live  metal  parts  of  arresters  placed 
side  by  side;  the  following  limiting  distances  are  suggested  for 
guidance  in  installing  lightning  arresters,  such  as  those  of  the 
horn  gap  type  where  large  arcs  may  be  formed  and  blown  or 
drawn  from  one  element  to  another.  These  distances  may  be 
reduced  if  suitable  partitions  are  provided  between  the  arresters. 


Potential  difference,  volts 

Separation,  in. 

11  000 

24 

22  000 

32 

33,000  
44,000  

42 
50 

66  000  ' 

66 

88  000 

84 

110,000  

108 

99.  Choke  Coils. — When  a  lightning  arrester  is  connected  be- 
tween line  and  ground  in  or  near  generating  or  substations  for  the 
purpose  of  providing  a  path  to  ground  for  high-frequency  surges, 
an  inductive  reactance  is  placed  in  series  with  the  apparatus  to 
be  protected.  This  reactance  must  not  be  so  great  as  to  cause  a 
serious  drop  in  pressure  when  carrying  the  normal  line  current, 
neither  must  it  be  so  small  as  to  allow  the  induced  charges 
travelling  along  the  line  to  pass  through  it  rather  than  jump  the  air 
gap  of  the  lightning  arrester.  This  reactance  usually  takes  the 
form  of  an  air-insulated  coil  of  copper  wire  or  rod,  supported  at 
each  end  on  a  suitable  insulator.  The  "hour  glass"  form  of  coil, 
in  which  the  diameter  of  the  turns  increases  from  the  center  to- 
ward both  ends,  is  mechanically  stiffer  than  a  cylindrical  coil, 
and  any  arc  that  might  be  started  between  adjacent  turns  has  a 
greater  tendency  to  clear  itself.  The  air  space  between  turns  is 
usually  from  %  in.  to  %  in.  Too  little  attention  has  been  given 


PROTECTION  AGAINST  LIGHTNING  193 

in  the  past  to  the  proper  design  and  proportioning  of  choke  coils 
for  use  in  conjunction  with  lightning  arresters.  It  has  sometimes 
been  argued  that,  except  for  the  drop  of  pressure  under  working 
conditions,  and  the  higher  cost,  there  is  no  objection  to  installing 
very  large  choke  coils  having  a  high  inductance.  This  argument 
is,  however,  incorrect,  except  for  the  special  case  in  which  some 
protection  against  surges  is  provided  on  the  machine  side  of  the 
reactance  in  addition  to  the  lightning  arresters  on  the  line  side. 
A  high  reactance  may  be  quite  satisfactory  if  it  is  merely  in- 
tended to  hold  back  high-frequency  currents  travelling  along  the 
line;  but  surges  may  originate  near  the  generators  or  transformers 
due  to  switching  operations  or  other  causes,  and  a  very  high 
reactance  between  the  electrical  plant  and  the  line  will  tend  to 
aggravate  the  effect  of  comparatively  low-frequency  surges 
which  might  otherwise  be  dissipated  in  the  line,  or  even  through 
the  lightning  arrester.  In  fact,  choke  coils  should  be  designed 
with  due  regard  to  the  apparatus  they  are  intended  to  protect, 
with  a  view  to  avoiding  the  building  up  of  high  voltages  at  the 
terminals  of  the  generating  plant  in  the  event  of  surges  being 
set  up  in  or  near  the  plant  itself.  When  the  lightning  arrester 
discharges,  it  does  not  follow  that  high-frequency  waves  do  not 
find  their  way  through  the  choke  coil  to  the  machines;  but  the 
inductance  of  the  choke  coil  will  lower  the  frequency  of  such 
waves;  or,  in  other  words,  will  reduce  the  steepness  of  the  wave 
front  to  such  an  extent  that  the  insulation  of  the  machines  will 
not  be  injured.  The  first  few  turns  of  a  transformer  or  generator 
winding  will  act  as  a  choke  coil  and  usually  prevent  damage  to 
the  turns  farther  removed  from  the  terminals;  but  they  are  liable 
themselves  to  suffer  injury,  as  the  charge  will  leap  across  the  in- 
sulation and  so  get  to  ground.  If  it  is  assumed  that  the  reactance 
of  the  first  six  turns  of  a  transformer  winding  is  sufficient  to  afford 
protection  to  the  seventh  and  subsequent  turns  of  the  winding, 
then  a  choke  coil  having  a  reactance  equal  to  that  of  the  six 
turns  of  transformer  winding  will  afford  the  necessary  protection 
to  the  transformer.  A  higher  reactance  in  series  is  unnecessary 
and  may  be  dangerous. 

The  tendency  among  engineers  is  apparently  toward  the  use 
of  choke  coils  of  too  great  reactance.  As  an  example  of  what 
appears  to  be  generally  sufficient  to  afford  reasonable  protection 
to  modern  machinery,  about  25  turns  of  copper  rod  wound  into  a 
coil  10  in.  in  diameter  may  be  used  on  voltages  from  10,000  to 

13 


194  ELECTRIC  POWER  TRANSMISSION 

25,000,  while  for  pressures  of  the  order  of  100,000  volts,  two  such 
coils  would  be  connected  in  series.  The  diameter  of  the  copper 
rod  would  depend  upon  the  current  to  be  carried;  but  it  is  best 
to  have  it  large  enough  in  all  cases  to  be  self-supporting,  although 
coils  wound  on  insulating  frames,  with  separating  pieces  between 
turns,  are  not  necessarily  objectionable. 

It  is  possible  that  copper  is  generally  used  for  choke  coils  be- 
cause the  calculation  of  the  reactance  at  various  frequencies 
is  more  easily  made  than  in  the  case  of  a  "magnetic"  material, 
such  as  iron;  but  the  cost  can  be  reduced  by  using  iron  bar  or 
strip  in  place  of  copper,  and  a  peculiarity  of  the  iron  choke  coil 
is  its  property  of  passing  currents  of  normal  frequency  with  com- 
paratively small  loss  of  pressure,  while  the  choking  effect  with 
high-frequency  currents  is  very  much  greater. 

100.  Arcing  Ground  Suppressor. — If  any  one  conductor  of 
a  transmission  system  is  connected  to  ground  through  an  arc 
such  as  might  occur  over  an  insulator  in  the  event  of  a  rise  of 
pressure  due  to  any  cause,  there  is  the  possibility  of  the  arc 
continuing  during  an  appreciable  length  of  time,  sufficient  to  do 
serious  damage  to  the  insulator,  even  if  it  should  not  totally 
destroy  it.  Apart  from  this  danger,  every  intermittent  arc  is 
liable  to  set  up  dangerous  high-frequency  surges  in  the  line, 
especially  at  the  moment  when  it  is  finally  interrupted.  To 
protect  a  line  against  troubles  due  to  this  cause,  a  device  known 
as  the  arcing  ground  suppressor  has  been  introduced.  This  is 
an  automatic  device  for  momentarily  short-circuiting  the  arc 
through  a  switch.  By  providing  a  metallic  connection  between 
the  conductor  and  ground,  the  arc  is  suppressed,  and  it  will 
usually  not  re-form  when  the  switch  is  again  opened,  because 
the  air  in  the  path  of  the  arc  has  had  time  to  cool,  and  the  line 
pressure,  which  was  sufficiently  high  to  maintain  the  arc  once 
started,  is  not  able  to  break  down  the  insulation  of  the  new  layers 
of  cooler  air.  The  arcing  ground  suppressor  was  fully  described  in 
the  Proceedings  A.  I.  E.  E.  of  March,  191 1,1  but  the  diagram,  Fig. 
72  will  explain  the  principle  of  its  action.  Automatic  switches 
are  provided  which  will  connect  any  one  conductor  to  ground  dur- 
ing the  very  short  time  necessary  to  allow  the  arc  to  clear  itself. 
The  principal  feature  of  the  device  is  the  selective  relay  which 

1  "Protection  of  Electrical  Transmission  Lines,"  by  E.  E.  F.  Creighton. 
Trans.  A.  I.  E.  E.,  vol.  xxx  p.  257.  Refer  also  to  article  by  R.  A.  Marvin  in 
the  General  Electric  Review,  March,  1913. 


PROTECTION  AGAINST  LIGHTNING 


195 


will  energize  the  solenoid  operating  the  switch  on  the  faulty 
line.  On  high-pressure  systems,  this  relay  may  be  of  the  elec- 
trostatic type,  generally  on  the  principle  of  the  electrostatic 
ground  indicator.  On  comparatively  low-pressure  transmissions, 
the  forces  would  be  too  low  to  operate  such  a  device  satisfactorily, 
and  recourse  is  then  had  to  an  electromagnetic  relay  worked 
through  transformers.  There  are  no  difficulties  or  new  prin- 
ciples involved  in  the  design  of  such  a  relay.  When  the  relay  op- 
erates, the  switch  between  line  and  ground  is  momentarily  closed. 
On  re-opening,  a  suitable  resistance  is  inserted  before  the  final 
break,  to  prevent  the  creation  of  oscillating  currents  in  the  line. 


FIG.  72. — Diagram  of  arcing  ground  suppressor. 

101.  General  Remarks  on  Lightning  Protection.— The  best 
means  to  adopt  for  the  protection  of  any  particular  line  or  portion 
of  a  line  against  lightning  disturbances  is  still  largely  a  matter  of 
conjecture,  but  by  the  exercise  of  sound  judgment,  an  experi- 
enced engineer  should  be  able  to  provide  reasonable  protection 
against  discontinuity  of  service  during  atmospheric  disturbances. 
There  are  many  devices  to  choose  from,  each  of  which  has  a  par- 
ticular field  of  usefulness.  It  is  probable  that,  in  a  few  years'  time, 
the  additional  information  on  this  subject  which  is  continually  be- 
ing accumulated,  will  lead  to  uniformity  in  the  protective  arrange- 
ments adopted  under  the  various  conditions  arising  in  practice. 


196  ELECTRIC  POWER  TRANSMISSION 

In  the  meanwhile,  however,  a  careful  record  of  all  accidents  due 
to  lightning  or  abnormal  pressure  rises  should  be  kept  in  connec- 
tion with  each  power  system  with  overhead  transmission,  as  this 
will  generally  lead,  after  careful  investigation,  to  certain  amplifi- 
cations or  modifications  of  the  existing  protective  arrangements 
such  as  to  prevent  the  repetition  of  similar  accidents.  In  this 
manner,  very  fair  protection  can  be  afforded  at  the  present  day 
to  almost  any  overhead  transmission  system;  but  it  is  doubtful 
if  it  will  ever  be  possible  to  protect  apparatus  against  a  direct 
lightning  stroke.  Damage  to  machinery,  due  to  this  cause,  is, 
however,  very  rare. 

In  regard  to  the  protection  of  the  line  itself,  it  is  obvious  that 
protective  devices,  however  complete  or  perfect  they  may  be, 
provided  at  the  two  ends  of  a  long  transmission,  afford  no  protec- 
tion to  the  insulators  along  the  line.  The  frequently  grounded 
guard  wire  would  appear  to  be  a  good  protection  to  a  line;  but  here 
again  the  engineer  must  use  his  judgment,  because  certain  por- 
tions of  a  line  may  require  far  more  protection  than  other  por- 
tions, and  even  if  the  cost  of  guard-wire  protection  be  considered 
excessive  for  the  entire  length  of  a  long-distance  transmission,  it 
may  yet  be  a  decided  advantage  to  provide  guard-wire  protection 
near  the  generating  and  transforming  stations  and  on  those  parts 
of  the  line  most  likely  to  be  affected  by  atmospheric  disturbances. 

In  some  cases,  it  may  be  wise  to  improve  the  insulation  and 
to  raise  the  voltage  at  which  a  "spill  over"  will  occur;  while 
under  other  circumstances  it  might  be  better  to  provide  an  easy 
path  for  a  discharge  over  insulators,  by  means  of  suitably  disposed 
arcing  rings  or  equivalent  arrangement.  Mr.  P.  H.  Thomas  once 
explained  the  matter  of  line  insulation  by  making  use  of  a  very 
simple  analogy.  Where  a  discharge  strikes  the  line,  a  wave  starts 
and  the  potential  of  this  wave  will  be  such  as  can  be  allowed 
by  the  line  itself;  the  energy  of  the  discharge  is  limited  by  the 
static  capacity  of  the  line  and  the  voltage  at  which  a  "spill 
over"  will  occur  at  the  insulators.  The  energy  of  the  travelling 
waves  "grows  less  and  less  as  they  proceed.  This  action  may  be 
likened  to  the  formation  of  a  wave  in  a  long,  narrow  trough 
with  high  sides  containing  water  and  normally  less  than  half  full, 
by  sudden  flooding  of  the  trough  by  a  large  quantity  of  water  at 
some  particular  point;  the  excess  water  spills  over  and  escapes 
from  the  trough  at  the  point  of  the  flooding,  but  there  is  still  a 
wave  started  in  each  direction  as  high  as  the  sides  of  the  trough 


PROTECTION  AGAINST  LIGHTNING  197 

will  permit;  this  passes  along  until  the  end  is  reached  or  the  energy 
of  the  wave  is  gradually  dissipated.  It  makes  no  difference  how 
much  water  is  thrown  into  the  trough,  there  can  be  a  wave  only 
as  high  as  the  sides  will  permit." 

One  point  that  is  sometimes  overlooked  is  the  effect  of  the  line 
current  on  pressure  disturbances.  The  disturbances  that  are  set 
up  by  switching  operations  or  by  power  arcs  following  a  light- 
ning discharge,  will  be  far  more  serious  when  the  current  is  large 
than  when  it  is  small.  This  is  one  reason  why  extra  high-tension 
transmissions  suffer  less  from  lightning  disturbances  than  moder- 
ate voltage  systems  on  which  the  current  is  often  larger.  It  is 
hardly  an  exaggeration  to  say  that  the  handling  of  heavy  cur- 
rents on  long  distance  transmissions  presents  more  engineering 
difficulties  than  insulation  problems  on  the  high-voltage  schemes. 

Very  high- voltage  transmission  lines  may,  indeed,  operate  satis- 
factorily without  lightning  protection,  especially  when  working 
at  pressures  near  the  critical  voltage  of  the  corona  formation,  and 
some  relief  to  high-pressure  energy  is  afforded  by  the  corona 
itself.  Low-pressure  lines,  working  at  about  10,000  volts  are 
usually  far  less  exposed  than  the  high-pressure  lines,  and  the 
low-pressure  lightning  arresters  are  rather  more  effective  than 
those  for  the  higher  pressures.  Such  lines  do  not  give  so  much 
trouble  as  those  working  at  voltages  between  30,000  and  80,000. 

It  will  be  gathered  from  the  foregoing  remarks  that  not  only  the 
pressure  of  transmission,  but  the  amount  of  power  transmitted, 
is  an  important  factor  in  the  problem  of  lightning  protection. 

Without  attempting  a  detailed  analysis  of  the  troubles  due  to 
switching  operations,  it  may  be  stated  that,  as  a  general  rule,  it 
will  be  best  to  energize  a  dead  line  of  considerable  length  by  first 
connecting  to  the  line  the  step-down  transformers  at  the  distant 
or  receiving  end,  and  then  switching  the  step-up  transformers  at 
the  generating  end  on  to  the  low-tension  bus-bars. 

It  is  possible  that  the  near  future  may  see  some  developments 
in  the  matter  of  facilitating  the  dissipation  of  high-frequency 
energy  in  the  line  itself,  with  the  object  of  rapidly  limiting  the 
amplitude  of  the  travelling  waves  and  the  distance  from  the 
center  of  disturbance  at  which  their  effects  can  be  of  practical 
account.  It  is  obvious  that  what  is  required  is  a  line  that  will 
transmit,  wrthout  undue  loss,  the  power  currents  at  normal  fre- 
quency, and  yet  afford  means  for  the  rapid  dissipation  of  high- 
frequency  energy.  Apart  from  the  property  peculiar  to  the 


198  ELECTRIC  POWER  TRANSMISSION 

corona,  which  leads  to  the  dissipation  of  energy  on  over-voltages, 
there  is  a  property  common  to  all  metallic  circuits  which  leads 
to  the  more  rapid  dissipation  of  high-frequency  than  of  low-fre- 
quency energy.  The  so-called  "skin  effect"  which  apparently 
increases  the  resistance  of  a  conductor  carrying  alternating  or 
fluctuating  currents,  owing  to  the  forcing  of  the  current  toward 
the  outside  portions  of  the  wire  at  the  higher  frequencies,  is 
clearly  of  value  in  limiting  the  distance  over  which  high-fre- 
quency disturbances  are  propagated.  By  covering  the  conductor 
with  a  thin  layer  of  high-resistance  metal,  astonishing  results  can 
be  obtained.  Experiments  made  with  wires  having  a  coating 
of  nickel  only  0.07  mm.  thick,  showed  that  the  resistance  offered 
to  currents  of  300,000  cycles  per  second  was  four  times  the  re- 
sistance offered  by  the  same  wires  without  the  coating  of  high 
resistance  metal.  This  was  referred  to  by  Mr.  Gino  Campos  at 
the  Turin  International  Electrical  Congress  of  1911. 

In  conclusion,  it  would  seem  that  much  may  be  accomplished 
by  a  careful  study  of  local  conditions,  and  by  the  intelligent 
selection  of  such  protective  apparatus  as  may  be  available.  On 
the  other  hand,  it  is  not  improbable  that  the  coming  years  will 
see  fewer  rather  than  more  protective  devices  than  are  now  used. 
Careful  design  of  the  line  as  a  whole,  with  the  provision  of  suitable 
choke  coils  at  the  ends,  or  the  special  insulation  of  the  end 
turns  of  the  transformer  windings  to  withstand  very  much 
higher  pressures  between  turns  than  the  remainder  of  the  high- 
tension  winding,  will  frequently  be  all  that  is  necessary.  Even 
if  the  installation  of  costly  apparatus  may  slightly  decrease  the 
risk  of  damage  by  lightning,  the  economic  aspects  of  the  problem 
should  be  carefully  considered  before  deciding  to  provide  the 
additional  protection.  On  low  and  medium  voltage  systems, 
the  extra  expenditure  on  a  few  hundred  feet  of  underground  cable 
between  the  terminals  of  the  overhead  line  and  the  transformers 
or  h.t.  bus-bars,  will  generally  be  justified;  while  an  improved 
system  of  grounded  guard  wires  near  the  terminals  of  extra 
high-pressure  lines  will  tend  to  keep  the  original  disturbance  at  a 
safe  distance  from  the  station  apparatus. 

For  comparatively  low  voltages  the  use  of  condensers  appears 
to  be  justified;  but  they  are  always  costly,  and  it  is  questionable 
whether  the  money  invested  in  them  might  not  with  advantage 
be  spent  on  improved  insulation,  especially  in  the  immediate 
vicinity  of  the  terminal  apparatus. 


CHAPTER  VII 

TRANSMISSION  OF  ENERGY  BY  UNDERGROUND 
CABLES 

102.  Introductory. — The  principal  use  of  underground  electric 
cables  is  in  connection  with  distributing  systems  in  cities,  at 
comparatively  low  pressures;  but  they  are  also  used,  to  a  limited 
extent,  for  the  transmission  of  energy  at  fairly  high  voltages. 
In  this  chapter,  underground  cables  will  be  considered  chiefly 
in  relation  to  straight  long-distance  transmission  of  energy,  and 
what  follows  will  therefore  treat  mainly  of  power  cables  for  the 
transmission  of  energy  at  high  voltages.  No  attempt  will  be 
made  to  deal  with  the  historical  aspect  of  the  subject,  or  with  the 
practical  considerations  connected  with  the  handling,  laying,  and 
jointing  of  underground  cables;  but  the  construction  engineer 
and  the  student  desiring  more  information  than  can  be  contained 
in  a  single  chapter,  are  referred  to  the  many  other  writings  on 
this  subject,  including  the  excellent  book  on  "Underground 
Transmission"  by  Mr.  E.  B.  Meyer.1 

Notwithstanding  the  higher  cost  of  underground  transmission, 
it  replaces  overhead  transmission — for  comparatively  short 
distances — in  many  cases  where  the  latter  system  is  not  suitable. 
High-tension  underground  cables  are  used  in  populous  districts 
where  overhead  construction  is  not  permissible  or  advisable. 
The  underground  cable  is  not  subject  to  damage  by  wind,  ice, 
or  thunderstorms,  and  the  danger  to  life  is  obviously  reduced  by 
placing  the  high-voltage  conductors  underground.  The  un- 
sightly appearance  of  poles  and  overhead  conductors  in  the 
neighborhood  of  cities  is  another  reason  for  putting  wires  under 
ground,  notwithstanding  the  increase  of  cost.  This  consideration 
has  more  weight  in  Europe  than  in  America  and  partly  accounts 
for  the  fact  that  Europe  is,  and  always  has  been,  somewhat  in 
advance  of  America  in  the  design  and  manufacture  of  under- 
ground electric  cables.  The  shorter  distance  of  transmission, 

1  "Underground  Transmission  and  Distribution,"  by  E.  B.  Meyer. 
McGraw-Hill  Book  Co.  (1916). 

199 


200  ELECTRIC  POWER  TRANSMISSION 

resulting  in  lower  economical  pressures  on  the  Old  World  systems, 
is  another  reason  why  underground  power  cables  are  more  exten- 
sively used  in  Europe  than  on  the  American  continent.  The 
writer  has  therefore  no  hesitation  in  using  data  and  other  infor- 
mation referring  mainly  to  modern  British  practice,  especially 
since  he  has  been  able  to  secure  the  collaboration  of  Mr.  C.  J. 
Beaver,  chief  engineer  to  Messrs.  W.  T.  Glover  and  Co.  of  Man- 
chester, England.  Mr.  Beaver  has  not  only  furnished  much  use- 
ful advice  and  material  for  this  chapter,  but  he  has  also  kindly 
consented  to  read  and  correct  the  manuscript. 

Even  when  it  is  inadvisable  to  transmit  by  underground  cable 
the  whole  distance  of  transmission,  sections  of  the  line  passing 
through  populous  districts  may  be  put  underground,  while  over- 
head conductors  are  used  in  the  open  country.  Also,  as  men- 
tioned in  the  preceding  chapter,  a  short  length  of  underground 
cable  is  desirable  as  a  protection  against  high-pressure  surges, 
at  each  end  of  a  long  overhead  transmission  line,  provided  the 
voltage  is  not  so  high  as  to  render  the  cost  prohibitive. 

Another  use  for  insulated  cables  is  as  feeders  on  electric  railway 
systems.  In  the  trunk-line  electrification  of  the  New  York 
Central  Railroad,  there  are  no  less  than  1,600,000  duct-feet  of 
conduit  for  insulated  feeder  cables,  some  of  which  are  of  tile, 
the  balance  consisting  of  iron  pipe. 

The  Thury  system  of  transmission  by  continuous  currents, 
which  is  explained  in  the  succeeding  chapter,  lends  itself  to  the 
extensive  use  of  underground  cables.  Smaller  and  cheaper 
cables  may  be  used  for  the  same  voltage  and  energy  transmitted 
on  a  D.  C.  than  on  an  A.  C.  transmission.  Mr.  J.  S.  Highfield 
has  shown1  that  if  the  overall  diameter  of  a  100,000  volt  single- 
core  lead-covered  cable  for  alternating  currents  is  3.27  in.,  the 
equivalent  cable  for  100,000  volts  continuous  current  would  have 
an  outside  diameter  of  only  1.75  in.  It  is  true  that  the  compari- 
son is  between  the  r.m.s.  values  and  not  the  maximum  values  of 
the  voltage;  but  it  serves  to  show  that  a  considerable  reduction 
of  size — and  therefore  of  cost — is  effected  when  cables  are  used 
on  D.C.  instead  of  A.  C.  systems. 

103.  Submarine  Power  Cables. — When  transmitting  electric 
energy  across  water  which  cannot  be  spanned  by  overhead 
conductors,  the  insulated  cable  becomes  a  necessity.  Two 

1  Discussion  of  Mr.  Beaver's  paper  on  "Cables,"  Jour.  Inst.  E.  E.,  vol. 
53,  Dec.  15,  1914  and  Mar.  1,  1915. 


TRANSMISSION  BY  UNDERGROUND  CABLES    201 

examples  of  submarine  power  transmission  occur  in  San  Fran- 
cisco, where  a  three-phase  12,000  -volt  cable  18,800  feet  long  has 
been  laid  across  the  bay,  and  two  11,000-volt  cables  have  been 
laid  across  the  Golden  Gate.  The  carrying  out  of  the  latter 
project  is  described  in  the  Electrical  World  of  March  4,  1916 
(Vol.  67,  No.  10,  p.  532).  The  greatest  depth  of  water  at  this 
point  is  210  feet.  The  cables  are  3-core,  each  13,500  feet  long, 
the  cores  being  insulated  with  rubber  (inside)  and  varnished 
cambric  (outside).  Impregnated  jute  is  used  as  a  filler.  The 
lead  sheath  is  ^2  in-  thick,  and  the  armoring  consists  of  42  No. 
4  B.W.G.  galvanized  steel  wires.  The  deep  water  portion  of 
the  cable  has  cores  of  250,000  circ.  mil  section,  with  an  outside 
diameter  of  4  in.,  and  a  weight  of  19  Ib.  per  foot.  At  the  shore 
ends,  the  cores  are  of  350,000  circ.  mil  section,  with  an  overall 
diameter  of  4^  in.  and  a  weight  per  foot  of  22  Ib. 

In  Germany  there  is  the  Stralsund  channel  project  transmitting 
three-phase  power  at  15,000  volts  from  the  generating  station 
at  Stralsund  to  the  island  of  Riigen,  the  total  length  of  submarine 
cable  being  5600  feet.  Another  example  occurs  in  the  straits 
between  the  mainland  and  the  island  of  Fehmern,  where  two 
3300-foot  lengths  of  lead-covered,  iron  wire  armored,  paper- 
insulated  11,000-volt  cables  are  laid  under  water. 

The  most  important  existing  submarine  power  transmission 
is  the  international  cable  between  Sweden  and  Denmark,  which 
was  put  into  operation  in  December,  1915.  It  is  3.4  miles  long 
and  connects  a  point  near  Palsjo  in  Sweden  with  Marienlyst 
in  Denmark.  The  cable  is  three-core  with  impregnated  paper 
insulation  suitable  for  a  working  pressure  of  35,000  volts  (test 
pressure  87,500  volts).  The  cross-section  of  each  core  is  0.108 
sq.  in.  and  steel  wire  armoring  is  applied  over  the  lead  sheath. 
The  greatest  depth  of  the  Oresund  at  the  point  traversed  by  the 
cable  does  not  exceed  125  feet.  The  cable  was  laid  in  nine  lengths 
of  about  2000  feet. 

104.  Voltage  Limitations  of  Underground  Cables.  —  Trans- 
mission and  distribution  by  underground  cables  with  alternating 
currents  at  20,000  and  22,000  volts  is  not  at  all  uncommon;  but 
there  are  not  many  instances  of  higher  voltage  cable  systems  at  the 
present  day.  Various  schemes  are  in  progress  or  in  contempla- 
tion in  England  for  linking  up  separate  systems  or  parts  of 
systems  at  pressures  from  30,000  to  60,000  volts.  A  few  years 
ago  an  extensive  system  of  30,000  volt  three-phase  cables  was 


TAT*  TSACM«W» 
NTA  BARBARA, 


202  ELECTRIC  POWER  TRANSMISSION 

laid  in  Berlin,  following  the  experience  gained  in  England  with 
20,000  volts.  There  are  a  few  short  lengths  of  40,000-volt  three- 
core  three-phase  cables  in  France,  and  this  is  probably  the  high- 
est working  voltage  applied  to  three-core  cables  at  the  present 
time.  The  difficulties — apart  from  high  cost — in  the  way  of  in- 
creasing the  pressure  limit,  include  heavy  capacity  currents  and 
large  dielectric  losses,  the  latter  being  "all  day"  losses  like  the 
iron  losses  of  distributing  transformers.  The  size,  or  diameter 
over  the  lead  sheathing,  must  not  be  overlooked;  a  high-tension 
cable  with  an  outside  diameter  much  in  excess  of  4  to  4)^  in. 
would  not  only  be  difficult  to  handle,  but  would  be  liable  to 
injury  while  being  laid  in  position  unless  precautions  were  taken  to 
avoid  sharp  bends  and  mechanical  injury  during  the  process  of 
laying. 

It  seems  nevertheless  probable  that  in  the  near  future  we 
shall  see  important  developments  in  high-voltage  cable  transmis- 
sion, notably  in  connection  with  electric  railways  in  Switzerland 
and  elsewhere  on  the  continent  of  Europe,  where  power  cables 
may  be  used  for  pressures  up  to  80,000  and  even  100,000  volts, 
and  of  sufficient  size  to  transmit  from  10,000  to  20,000  k.v.a. 
per  cable.  In  this  connection  it  should  perhaps  be  pointed  out 
that  the  much  advertised  60, 000- volt  underground  transmission 
for  supplying  single-phase  power  to  the  Dessau-Bitterfeld  rail- 
road in  Germany  from  the  power  station  at  Muldenstein,  employs 
30,000-volt  single-core  cables,  the  middle  point  of  the  60,000-volt 
system  being  grounded  through  a  resistance  at  the  generating 
station.  The  use  of  single-core  cables  in  this  manner  entails  the 
use  of  costly  methods  of  laying,  because  it  precludes  armoring 
(owing  to  the  excessive  losses  that  would  occur),  and  necessi- 
tates insulation  between  the  lead  sheaths  and  ground.  The 
feature  of  chief  interest  from  the  cable  maker's  point  of  view  is 
the  employment  in  these  cables  of  aluminum  in  place  of  copper 
conductors,  the  object  being  to  reduce  the  voltage  gradient  at  the 
surface  of  the  conductor.  This  point  will  be  referred  to  again 
later. 

105.  Types  and  Construction  of  Power  Cables. — For  the  trans- 
mission of  three-phase  alternating  currents,  three-core  cables 
may  be  used  for  pressures  up  to  about  60,000  volts.  When  the 
pressure  exceeds  this  figure,  a  separate  single-core  cable  should 
be  used  for  each  phase.  These  separate  cables  of  an  extra  high- 
tension  three-phase  transmission  should  be  symmetrically  ar- 


TRANSMISSION  BY  UNDERGROUND  CABLES    203 

ranged  with  a  small  amount  of  insulation  between  the  lead  sheath 
and  ground.  Steel  armored  single-core  cables  are  inadmissible 
for  alternating  currents  because  of  the  inductive  effects. 

The  lead  sheathing  referred  to  is  a  necessity  because  impreg- 
nated paper  is  practically  the  only  material  at  present  available 
for  the  insulation  of  e.h.t.  underground  cables.  It  is  somewhat 
hygroscopic  in  character,  and  the  protection  of  the  lead  covering 
is  therefore  provided.  Fig.  73  shows  a  section  through  a  three- 
core  30,000-volt  paper-insulated  underground  cable,  while  Fig. 
74  shows  one  of  three  separate  cables  designed  for  a  working 
pressure  of  173,000  volts  between  phases.1  This  cable  is  provided 


FIG.  73. — Three-core  30  kv.  lead  covered  cable. 

with  a  so-called  "inter-sheath"  of  lead,  the  purpose  of  which 
is  to  improve  the  design  by  reducing  the  potential  gradient  at 
the  surface  of  the  conductor.  The  theory  of  this  method  will  be 
discussed  later. 

The  "clover  leaf"  or  sector  type  of  three-core  cable  is  illus- 
trated in  Fig.  75.  This  design  permits  of  a  smaller  overall 
diameter  for  a  given  voltage  and  so  leads  to  a  saving  in  cost; 
but  it  is  rarely  used  for  pressures  exceeding  20,000  volts.  The 
sector  shape  of  cross-section  is  obtained  by  specially  stranding 
wires  of  suitably  varied  diameters,  or  by  rolling  or  hammering 
a  circular  strand  to  the  desired  shape.  This  construction  was 
introduced  in  Europe  many  years  before  it  was  adopted  in 
America;  but  British  practice  does  not  favor  the  use  of  sector- 

1  From  Mr.  C.  J.  Beaver's  paper  on  "Cables"  in  the  Jour.  Inst.  E.  E., 
vol.  53,  Dec.  15,  1914. 


204 


ELECTRIC  POWER  TRANSMISSION 


shaped  cores  in  three-phase  cables  for  working  pressures  exceeding 
11,000  or  12,000  volts. 
Concentric  cables  are  used  for  single-phase  transmissions  and 


Radial  thickness.    Radius. 

Conductor. — Inner  lead  tube — bore  0.27  in 0.060  in.  0.195  in. 

Copper  wires  19/15  8. W.G 0.072    '  0.267 

Outer  lead  tube 0.050    '  0.317 

Dielectric. — First  paper  layer 0.545    '  0.862 

Lead  inter-sheath 0.050    '  0.912 

Second  paper  layer 0.565    '  1.477 

Outer  sheath. — Lead  covering 0.160    '  1.637 

Complete  diameter 3.27  in. 

FIG.  74. — Section   through  single-phase  173  kv.  paper-insulated  cable,  with 
lead  "inter-sheath." 

are  therefore  practically  confined  to  railway  work.  Fig.  76 
is  a  section  through  such  a  cable,  the  return  conductor  being  in 
the  form  of  a  layer  of  segmental  strips  laid  over  the  insulation  sur- 


Lead 


Paper 


FIG.  75. — Three-core  cable  with  shaped  cores. 

rounding  the  inner  core.  When  this  outer  conductor  is  grounded 
— as  it  usually  is — the  arrangement  is  very  satisfactory  and 
efficient.  Concentric  cables  are  simple  to  construct,  and  even 


TRANSMISSION  BY  UNDERGROUND  CABLES    205 

when  used  with  alternating  currents,  they  can  be  armored  and 
therefore  buried  directly  in  the  ground,  thus  facilitating  the  dis- 
sipation of  heat  which  is  easily  carried  from  the  outer  conductor 
to  the  lead  sheath  and  armor. 

There  are  two  methods  of  manufacturing  paper-insulated 
cables.  In  both  methods  a  roll  of  paper  is  cut  into  narrow  rib- 
bons which  are  lapped  on  the  copper  conductor  in  successive 
layers  until  the  required  thickness  is  attained.  In  one  method 
the  paper  is  put  on  dry  and  the  whole  cable  is  immersed  in  the 
insulating  oil.  In  the  other,  the  impregnation  is  effected  by 
passing  the  paper,  before  it  is  cut  into  ribbons,  through  a  bath 
of  the  insulating  oil.  This  last  method,  which  was  evolved  and 


FIG.  76. — Section    through    concentric    cable   for   single-phase   transmission. 

has  been  used  for  the  last  twenty  years  by  Messrs.  W.  T.  Glover 
and  Co.  of  Manchester,  England,  appears  to  have  the  advantage 
of  more  certain  and  thorough  impregnation  of  the  insulation. 

With  paper  insulation  the  lead  sheath  must  be  very  carefully 
applied  to  ensure  the  absence  of  flaws  and  "pin  holes"  through 
which  moisture  might  be  admitted.  The  lead  sheathing  is 
applied  by  passing  the  insulated  cable  through  dies  in  a  hydraulic 
press  which  forces  hot  lead  in  a  semi-fluid  condition  around  the 
slowly  moving  cable,  thus  forming  a  closely  fitting  lead  cylinder 
on  the  outside  of  the  insulation.  It  is  the  practice  of  one  English 
factory  to  apply  a  hydraulic  test  to  the  finished  cables  in  order 
to  detect  the  presence  of  imperfections  in  the  lead  covering, 
by  forcing  water  into  the  dielectric  so  that  its  presence  may  be 
detected  before  the  cable  leaves  the  factory.  It  is  partly  in  order 


206  ELECTRIC  POWER  TRANSMISSION 

to  facilitate  such  testing  that  the  same  firm  employs  a  metallic 
test  sheath1  enclosed  in  the  cable,  and  lightly  insulated  from  the 
lead  covering.  This  lead  sheath  is  also  useful  for  maintenance 
tests,  fault  localization,  the  detection  of  incipient  faults  on 
live  cables,  and  for  use  in  connection  with  special  protective 
devices  (Br.  Pat.  13681/17  Beaver,  Richards,  and  Claremont.)2 

Insulating  Materials  Other  Than  Insulated  Paper. — The  reason 
why  reference  has  not  previously  been  made  to  vulcanized 
rubber  and  vulcanized  bitumen  as  materials  for  cable  insulation 
is  not  because  these  are  not  in  general  use,  but  because  they  are 
not  so  suitable  as  paper  for  e.h.t.  power  cables.  Vulcanized 
bitumen  as  an  insulator  appears  to  have  met  with  greater  success 
in  Europe  than  in  America.  It  is  a  substance  of  which  the 
physical  properties  are  somewhat  similar  to  those  of  vulcanized 
rubber.  Cables  insulated  with  vulcanized  bitumen  (without 
lead  sheathing)  have  a  special  field  of  utility  in  mining  work, 
where  they  are  used  not  only  as  feeders  to  carry  the  electric 
energy  down  the  shaft,  but  also  as  distributors  under  ground. 
The  results  of  a  very  complete  investigation  of  the  properties  of 
this  material  will  be  found  in  Mr.  Beaver's  Institution  Paper 
on  "Cables"  previously  referred  to. 

A  type  of  insulation  which  is  largely  used  in  America  is  var- 
nished cambric,  or  varnished  cloth.  The  prepared  cloth  is 
applied  to  the  conductor  in  the  form  of  tape,  a  thin  layer  of  a 
non-hardening  viscous  filler  being  applied  between  the  layers  to 
ensure  flexibility  by  permitting  relative  movement  of  the  layers. 
Since  it  is  customary  to  provide  a  lead  sheath  over  power  cables 
insulated  with  varnished  cambric,  there  is  no  economic  advantage 
in  using  this  material  as  a  substitute  for  paper. 

106.  Methods  of  Laying  Underground  Cables.— The  present- 
day  methods  of  laying  underground  cables  may  be  classified 
under  three  headings: 

1.  Laid  direct  in  the  ground  (armored  cables). 

2.  Drawn  into  weldless  steel  tubes,   stoneware  ducts,   or 
fiber  ducts  in  concrete. 

3.  Laid  solid  in  wood,  cast  iron,  or  special  asphalt  troughing 
(this  last  is  a  patented  system  known  as  the   Howard 
asphalt  troughing). 

1  British  patent  No.  22355/12,  Beaver  and  Claremont. 

2  Particulars  of  the  test  sheath  cable  and  its  potentialities  will  be  found 
in  The  Electrician  (London)  of  July  5,  1918. 


TRANSMISSION  BY  UNDERGROUND  CABLES    207 

The  first-mentioned  system  is  usually  adopted  for  cross-country 
runs;  the  second,  in  congested  areas  where  streets  cannot  be 
disturbed  and  where  facilities  for  adding  to  the  number  of  cables 
are  desired.  The  third  system  may  be  adopted  where  occasional 
disturbance  of  the  street  is  not  a  matter  of  great  importance, 
or  where  chemical  protection  for  the  cable  is  desirable  on  account 
of  soil  conditions  or  other  causes. 

When  lead-covered  and  armored  cables  are  laid  direct  in  the 
ground,  it  is  customary  to  place  a  wooden  board  over  the  cable 
to  act  as  a  warning  to  workmen  and  so  protect  the  cable  in  the 
event  of  the  ground  surface  being  disturbed  after  the  cable  has 
been  laid. 

The  dra wing-in  system  is  the  one  most  commonly  adopted  at 
the  present  time.  The  ducts  for  the  cables  may  be  tile  or  fiber 
set  in  concrete,  or  they  may  consist  of  wrought-iron  pipes. 

Tile  or  stoneware  conduits  can  be  supplied  single-way  or 
multiple-way.  The  opening  is  usually  33^  in.  square  when 
used  for  distributing  purposes  in  or  near  cities.  The  inside 
corners  are  well  rounded,  and  the  outside  dimensions  are  approxi- 
mately as  follows: 

Single-way  duct 5  in.  by    5  in.  by  18  in.  long. 

Two-way  duct 5  in.  by    9  in.  by  24  in.  long. 

Three-way  duct 5  in.  by  13  in.  by  24  in.  long. 

Four-way  duct 9  in.  by    9  in.  by  36  in.  long. 

Six-way  duct 9  in.  by  1 3  in.  by  36  in.  long. 

These  ducts  are  set  in  concrete  forming  a  wall  about  3  in.  thick 
on  all  four  sides. 

Fiber  conduits  are  light  in  weight  and  easy  to  handle.  They 
are  cylindrical  pipes  made  of  wood  pulp  saturated  with  an  asphalt 
or  bituminous  compound  containing  a  small  amount  of  creosote, 
and  are  usually  supplied  in  5-ft.  lengths. 

Iron  pipes  are  useful  when  slight  bends  to  clear  obstructions 
are  of  frequent  occurrence.  These  can  be  supplied  in  20-ft. 
lengths  with  threaded  ends  and  couplings. 

In  all  cases,  the  internal  width  of  the  duct  should  allow  not  less 
than  ^-in.  clearance  for  the  largest  cable  to  be  drawn  into  it. 

Manholes  must  be  provided  at  intervals  of  300  to  500  feet, 
the  latter  distance  being  rarely  exceeded,  as  it  represents  the. 
practical  length  of  cable  of  large  size  which  can  be  drawn  into  an 
underground  duct  without  injury.  The  manholes,  if  required 


208  ELECTRIC  POWER  TRANSMISSION 

merely  for  dra wing-in  purposes  and  for  joints  (apart  from  the 
installation  of  transformers  or  other  apparatus),  should  measure 
about  7  ft.  X  6  ft.  X  6  ft.  high.  If  the  construction  permits 
of  an  arched  roof,  the  walls  might  be  about  5  ft.  high  with  a 
total  head  room  at  center  of  about  7  ft.  The  floor  should  be 
above  the  sewer  level,  and  proper  arrangements  provided  for 
drainage  and  ventilation.  If  transformers  are  placed  in  man- 
holes, a  space  of  about  4.5  cubic  feet  per  k.v.a.  should  be  provided. 


FIG.  77. — Section  through  three-core  cable  laid  in  Howard  asphalt  troughing. 

Fig.  77  is  a  section  showing  cables  in  the  Howard  asphalt 
troughing.1  The  chief  advantages  claimed  for  this  system  are 
imperviousness,  chemical  inertness,  ductility,  ease  of  repair, 
electric  insulation,  and  excellent  dissipation  of  heat  by  conduc- 
tion. In  the  latter  respect  it  has  the  advantage  over  all  other 
troughing  systems,  and  compares  favorably  with  armored 
cables  laid  directly  in  the  ground.2 

107.  Costs  of  Underground  Transmission  Lines. — The  cost  of 
•underground  systems  of  transmission  varies  greatly  with  the 

^British  patents  5945,  5946,  5947/06,  Stratton,  Claremont,  Beaver, 
Tanner. 

2  See  Beaver,  Jour.  I.  E.  E.,  vol.  47,  p.  747  (1911). 


TRANSMISSION  BY  UNDERGROUND  CABLES    209 

type  of  construction,  the  nature  of  the  ground,  the  road  surface, 
conditions  of  transport,  labor  facilities,  etc.;  but  the  following 
figures  should  be  useful  as  a  rough  indication  of  probable  cost, 
and  they  may  be  used  for  preliminary  estimates  when  reliable 
figures,  based  on  the  specific  conditions  and  requirements, 
are  not  available. 

Cost  per 


1.  Laid  direct. 

Cable  laid  direct  in  the  ground  and  covered  with  a  tarred 

warning  board $2200 . 00 

2.  Drawn  in. 

Cable  drawn  into  weldless  steel  tubes $6000.00 

Cable  drawn  into  stoneware  ducts $3200 . 00 

Cable  drawn  into  fiber  tubes  laid  in  concrete $4500.00 

3.  Laid  solid. 

Cable  laid  solid  in  Howard  Asphalt  trough,  complete 

with  bitumen  and  asphaltic  concrete $4200 . 00 

Cable  laid  solid  in  wood  troughing  with  Wood  covers, 
complete  with  bitumen $4500.00 

Cable  laid  solid  in  cast  iron  troughing  with  tile  covers, 
complete  with  bitumen $6200.00 

The  troughing  and  ducts  above  referred  to  allow  for  suitable 
clearances  for  cables  of  about  4  in.  diameter.  These  costs  are 
based  on  the  assumption  that  not  less  than  six  cables  are  laid  at 
a  time  in  the  case  of  systems  (1)  and  (3),  and  not  less  than  twelve 
duct-ways  in  the  case  of  (2). 

In  order  to  arrive  at  the  total  cost  of  a  given  system  of  under- 
ground transmission,  the  cost  of  cables  must  be  added  to  that 
of  the  conduit.  It  is  obviously  impossible  to  give  close  prices, 
and  quotations  for  cables  should  always,  if  possible,  be  obtained 
from  the  manufacturers;  but  the  prices  here  given  will  serve  as  a 
rough  indication  which  will  often  determine  whether  or  not  it 
will  be  worth  while  proceeding  further  in  the  matter  of  a  proposed 
undertaking. 

The  ducts  and  troughs  for  which  costs  have  been  given  would 
be  capable  of  accommodating  cables  of  the  approximate  carrying 
capacities  as  listed  in  the  following  table. 

The  expression  "voltage  graded"  as  here  used  refers  to  the 
system  of  providing  metallic  layers  between  sections  of  the 
insulation  with  a  view  to  economy  of  size  and  cost,  in  the 
manner  which  will  be  explained  later. 

14 


210 


ELECTRIC  POWER  TRANSMISSION, 


Approxi- 

Approxi- 

System 

Type  of  cable 

Working  pressure, 
k.v. 

Kilowatts 

mate  cost 
per  mile 

mate  cost 
per  mile 

per  cable 

per  kw. 

3-phase 

3-core 

33 

15,000 

$12,000 

$0.80 

3-phase 

3-core 

66 

20,000 

13,500 

0.675 

3-phase 

Single  core 

175    per    phase; 

40,000 

15,000 

0.375 

100   to   ground 

per  core 

(voltage-graded  ) 

Single  phase 

Concentric 

100 

30,000 

12,000 

0.40 

(grounded 

(voltage-graded  ) 

outer) 

NOTE. — Single  wire  armoring  and  jute  serving  for  laying  direct  in  the 
ground  would  add  approximately  30  per  cent,  to  the  above  figures  of  cost. 


SKETCH  A 

Cable  Moulding 

D  Diameter  Depends  on 

Size  at  Cable  Installed 


loM 


FIG.  78. — Junction  between  underground  and  overhead  conductors. 


TRANSMISSION  BY  UNDERGROUND  CABLES    211 

108.-  Cable  Terminals.    Junction  with  Overhead  Lines. — The 

connection  between  overhead  wires  and  underground  cables  must 
be  made  with  great  care  to  avoid  trouble  at  this  point.  The 
ends  to  be  attained  and  the  precautions  to  be  taken  are  fairly 
obvious,  and  the  engineering  difficulties  may  readily  be  overcome. 
Each  particular  case  should  be  considered  as  a  special  problem, 


•Bnbber  Gasket 


Fio.  79. — Detail  of  cable  ends  at  junction  with  overhead  line. 

and  the  proper  steps  taken  to  provide  adequate  insulation  and 
prevent  deterioration  or  damage  by  water  owing  to  improperly 
sealed  joints. 

Fig.  78  shows  the  terminal  pole  of  an  overhead  line  and  the 
methods  employed  for  the  protection  of  the  cable.  It  is  repro- 
duced by  kind  permission  of  Mr.  E.  B.  Meyer  and  his  publishers 
from  "Underground  Transmission  and  Distribution."  The 
design  of  the  actual  terminals  or  "pot-heads"  as  they  are  some- 
times named,  varies  considerably.  One  form — suitable  for  a  work- 
ing pressure  of  11,000  volts,  three  phase — is  shown  in  Fig.  79 


212  ELECTRIC  POWER  TRANSMISSION 

which  has  been  reproduced  from  a  drawing  kindly  supplied  by 
Messrs.  W.  T.  Glover  &  Co.  Ltd.,  Manchester,  England. 

109.  Design  of  Cables.  —  The  theory  of  the  Potential  Gradient 
at  the  surface  of  a  wire  of  radius  r  surrounded  by  a  concentric 
metallic  cylinder  of  internal  radius  R  has  already  been  discussed 
in  Article  76  of  Chapter  V,  in  connection  with  insulating  bushings. 
With  a  dielectric  of  constant  specific  inductive  capacity  through- 
out the  total  thickness,  the  maximum  potential  gradient  —  which 
occurs  at  the  surface  of  the  inner  conductor  —  is  given  by  formula 

(61); 

77T 

G  =  —     —  p  volts  per  centimeter  (61) 

i       " 
r  log,  - 

where  E  stands  for  the  potential  difference  between  the  conductor 
and  the  outside  metallic  sheath,  in  volts,  and  the  dimensions  are 
expressed  in  centimeters.  This  formula  is  therefore  applicable 
to  single-core  cables  with  lead  sheath,  and  to  the  concentric  type 
of  two-conductor  cable. 

The  breakdown  gradient  of  paper-insulated  cables  is  about 
200  k.v.  per  centimeter,  and  it  is  advisable  not  to  exceed  a  figure 
of  120  to  150  k.v.  for  the  guaranteed  test  pressure.  Since  the 
test  pressure  may  be  as  high  as  2^  times  the  working  pressure, 
the  maximum  working  stress  will  be  of  the  order  of  50  k.v.  per 
centimeter.  It  is  quite  possible  to  design  practical  paper- 
insulated  cables  for  working  stresses  of  60  to  80  k.v.  per  cm.; 
but  owing  to  the  fact  that  dielectric  heating  is  likely  to  be  exces- 
sive with  these  high  gradients,  from  40  to  50  k.v.  per  cm.  is  a 
reasonable  maximum.  All  the  voltages  here  referred  to  are 
r.m.s.  values,  and  not  peak  values. 

The  Capacity  of  a  single-core  cable  per  centimeter  of  length  is 
the  reciprocal  of  the  elastance  as  given  by  formula  (59),  whence 

Capacity  per  centimeter  =  --  ^  farads  (79) 


where  k  is  the  relative  inductive  capacity  or  dielectric  constant 
of  the  insulating  material,  the  value  of  k  for  air  being  unity. 
The  numerical  value  of  K,  as  given  in  Article  72  of  Chapter  V, 
is  8.84  X  10~14,  and  since  there  are  161,000  centimeters  in  a 
mile,  the  capacity  per  mile  of  single-core  cable,  in  microfarads,  is 


TRANSMISSION  BY  UNDERGROUND  CABLES    213 
161000  X  27r  X  8.84  X  106  X  k 


10"  X  log, 
k 


=  0.0896 


=  0.0388 -—  (80) 


Approximate  values  of  k  which  apply  to  materials  used  for 
cable  insulation  are: 

k  =  3.3  for  impregnated  paper. 
k  =  4.2  for  varnished  cambric. 
k  =  4.5  for  vulcanized  bitumen. 
k  =  3  to  6  for  vulcanized  india-rubber. 

The  permittivity,  or  relative  specific  inductive  capacity  (k) 
will  depend  somewhat  upon  the  temperature  of  the  dielectric. 
The  effect  of  the  higher  temperatures  in  increasing  the  value  of 
k  being  greatest  for  varnished  cambric  insulation.  The  above 
values  are  averages  for  normal  working  temperatures. 

The  formula  for  the  Insulation  Resistance  of  •  a  single-core 
cable  is  obtained  as  follows: 

Let  p  stand  for  the  resistivity,  or  specific  resistance,  of  the 
dielectric,  in  megohms.  It  is  the  electrical  resistance  of  one 
centimeter  cube  of  the  cable  insulation.  Considering  the  insula- 
tion resistance  of  a  length  of  cable  I  centimeters  long  to  be  the- 
sum  of  the  resistances  of  successive  concentric  layers  of  insula- 
tion, we  may  write, 

dx 


p 


2-n-xl 


where  dx  is  the  length,  and   (2-^x1)  the  area,  of  a  cylindrical 
element  of  radius  x  and  length  I.     Whence 


" 

aS  J 


Let  R{  stand  for  the  insulation  resistance,  in  megohms,  of  one 
mile  of  cable;  then  I  =  161,000  cm.  ;  and  if  we  convert  the  natural 
into  common  logarithms,  we  get: 


214  ELECTRIC  POWER  TRANSMISSION 

Ri  (of  one  mile)  =  2.28  X  10~6  X  p  Iog10  (— )  megohms  (82) 

Some  average  values  of  p  are : 

For  impregnated  paper,  p  =  8  X  108 

For  varnished  cambric,  p  =  3  X  108  to  5  X  108 

For  vulcanized  bitumen,  p  =  2.4  X  108 

f  4.8  X  108 
For  vulcanized  india-rubber,  p  =  <         to 

( 1.2  X  1010 

Temperature  has  a  marked  effect  on  the  specific  resistance  of 
cable  insulation,  which  decreases  rapidly  with  increasing  tempera- 
tures. This  effect  is  particularly  noticeable  with  paper  insula- 
tion. Vulcanized  india-rubber  is  the  material  with  the  best 
temperature  characteristics. 

The  Reactance  of  a  single-conductor  cable  without  steel 
armoring  would  be  calculated  as  for  bare  wires- (refer  to  Articles 
43,  44,  and  45  of  Chapter  IV)  and  will  depend  upon  the  position 
of  the  return  conductor.  Single  cables  carrying  alternating 
currents  should  not  be  armored  or  drawn  into  iron  pipes. 

The  reactance  of  two-conductor  concentric  cables  is  usually 
negligible.  Formulas  for  three-core  cables  are  referred  to  in 
Article  112. 

110.  Economical  Core  Diameter  of  High-pressure  Cables. — 
For  a  given  voltage  and  constant  diameter  over  the  insulation 
of  a  single-core  cable,  there  is  a  definite  diameter  of  the  core 
which  will  cause  the  potential  gradient  at  the  conductor  surface 
to  be  a  minimum.  A  small  core  will  allow  of  a  greater  thickness 
of  insulation;  but  on  the  other  hand  the  smaller  radius  of  curva- 
ture will  tend  to  increase  the  stress;  while  the  effect  of  too  large 
a  diameter  of  core  is  also  to  cause  an  increase  in  the  stress  through 
reduction  of  the  total  thickness  of  insulation.  Formula  (61) 
may  be  written, 

E 


and  if  we  assume  both  R  and  G  to  be  constant,  the  greatest  maxi- 
mum permissible  voltage  (E)  will  be  obtained  when  the  quantity 

r  l°g«  (~)    is  a  maximum,  a  condition  which  is  satisfied  when 
log*  (j  =  1,  or  R  =2. 718  r.     Assuming  this  ideal  requirement  to 


TRANSMISSION  BY  UNDERGROUND  CABLES    215 

be  fulfilled,  even  if  the  inner  core  has  to  be  made  hollow  in  order 
to  economize  conductor  material,  we  obtain  the  relation: 

r-%  (83) 

from  which  an  approximate  value  of  economic  conductor  radius 
can  be  obtained.  This  radius  will  generally  (in  high  voltage 
cables)  be  found  to  be  greater  than  that  required  to  provide 
the  necessary  cross-section  for  current  carrying  purposes,  and 
this  accounts  for  the  fact  that  aluminum  may  prove  to  be  more 
economical  than  copper  as  a  conductor  in  high-tension  under- 
ground cables.  The  30,000-volt  cables  for  the  electrification  of 
the  Dessau-Bitterfeld  railway,  previously  referred  to,  have  cores 
of  stranded  aluminum  0.512  in.  diameter  (0.157  sq.  in.  cross-sec- 


Inner  lead  tube  ...........  0.256  in.  bore,  or  0.140  in.  radial. 

Conductor  ...............  24/0.082  in. 

Outer  lead  sheath  .........  0.05  in.  radial. 

Complete  diameter  ........  0.80  in. 

FIG.  80.  —  Specially  constructed  core  for  e.h.t.  underground  cables. 

tion)  covered  with  impregnated  paper  of  a  radial  thickness  of 
0.512  in.     The  lead  sheath  is  0.138  in.  thick  and  it  is  covered 
with  a  layer  of  jute,  the  overall  diameter  of  the  cable  being  2.05 
in.     By  formula  (61)  the  maximum  gradient  is 
__  30000 


=  42,000  volts  per  centimeter. 

If  stranded  copper  had  been  used  (without  a  hollow,  or  non- 
metallic  core)  the  diameter  of  the  core  of  equivalent  current-carry- 
ing capacity  would  have  been  about  %  in.  which,  with  the  same 
thickness  of  insulation,  would  cause  the  maximum  gradient  to 
be  about  48,000  volts  per  centimeter. 

The  application  of  formula  (61)  to  the  calculation  of  the  maxi- 
mum potential  gradient  in  cables  with  stranded  wire  cores, 


216  ELECTRIC  POWER  TRANSMISSION 

gives  only  approximate  results.  The  comparatively  small  radius 
of  the  individual  wires  will  bring  about  a  concentration  of  stress 
appreciably  greater  than  what  would  be  indicated  by  formula  (61). 
One  method  of  providing  a  smooth  cylindrical  surface  to  the  cable 
core  is  to  sheath  over  the  stranded  copper  core  with  a  thin  wall 
of  lead.  Fig.  80  shows  the  complete  core  (without  insulation) 
of  a  high-pressure  cable  as  designed  by  a  British  cable  factory.1 
In  this  case  the  calculated  "economic"  radius  for  a  maximum 
gradient  of  50,000  volts  (r.m.s.)  is  0.8  in.  and  since  the  conductor 
is  of  sufficient  carrying  capacity  when  made  up  of  24  copper  wires 
of  0.082  in.  diameter,  these  wires  are  disposed  around  a  hollow 
lead  core  of  the  necessary  diameter,  as  shown  in  the  illustration. 
111.  Grading  of  Cables. — The  potential  gradient  at  any  dis- 
tance x  from  the  center  of  a  single-core  cylindrical  cable  is  given 
by  formula  (58)  of  Article  76,  Chapter  V,  as 


2irxKk 

For  a  given  voltage  and  total  capacity  of  the  cable — which 
determine  the  total  dielectric  flux  SF — it  follows  that  if  k  is  also 
constant,  the  gradient  will  have  a  value  inversely  proportional 
to  the  distance  (x)  from  the  center  of  the  core.  With  a  view  to 
reducing  the  outside  diameter  of  the  cable,  the  value  of  k  should 
change  continually  in  successive  layers  of  insulation  in  order  to 
maintain  G  as  nearly  as  possible  at  the  maximum  permissible 
value  throughout  the  thickness  of  the  insulation.  In  other  words, 

if  k  a  -  the  potential  gradient  will  be  constant  at  all  parts  of  the 

dielectric.  By  varying  the  nature  of  insulation  in  adj  acent  layers, 
a  practical  approximation  to  this  condition  can  be  obtained 
— especially  with  rubber,  which  can  be  made  in  varying  qualities 
having  markedly  different  permittivities.  It  cannot  be  said, 
however,  that  the  grading  of  cables  by  this  method — i.e,  by  con- 
trolling the  capacities  of  successive  thicknesses  of  insulation — 
has  met  with  much  success  in  practice,  although  developments 
along  these  lines  may  be  looked  for  when  the  higher  voltages 
for  underground  transmission  become  more  common. 

Another  method  of  grading  'cables,  which  might  be  described  as 
the  conducting  layer  method,  gives  promise  of  coming  to  the  front 
as  soon  as  very  high  voltage  cables  are  in  greater  demand  than 

1  British  patent  No.  20549/14. 


TRANSMISSION  BY  UNDERGROUND  CABLES    217 

they  are  at  present.  The  principle  is  somewhat  similar  to  that 
of  the  condenser  type  of  bushing  (see  Article  77,  Chapter  V) 
except  that  the  intermediate  metallic  cylinders  are  now  all  of 
the  same  length,  and  the  potential  gradient  is  controlled  by 
"anchoring"  the  voltage  of  these  metallic  intersheaths.  The 
term  "voltage  grading"  might  be  used  to  distinguish  this  method 
from  "  capacity  grading."  The  insulation  is  the  same  throughout 
the  total  thickness,  but  being  divided  into  two  or  more  sections 
by  means  of  metallic  cylinders,  each  section  can  be  made  to  take 
its  proper  share  of  the  total  potential  difference  by  applying  a 
definite  voltage  from  an  outside  source  to  the  intermediate  me- 
tallic sheaths.  Various  methods  of  accomplishing  the  necessary 
distribution  and  "  anchoring  "  of  potential  for  both  D.  C.  and  A.  C. 
cables  have  been  patented  by  Tanner  and  Claremont.1  The 
most  obvious  way  of  obtaining  the  required  voltage  control  on 
an  A.  C.  system  is  to  take  tappings  from  the  high  voltage  side 
of  the  power  transformers  for  connection  to  the  intermediate 
sheaths. 

112.  Three-core  Cables.— The  formulas  for  use  with  three- 
core  cables  are  not  so  easily  developed  as  those  for  single-core 
cables,  and  they  are  largely  empirical,  especially  when  the  shape 
of  the  cores  departs  from  the  true  circular  section.     Tables  giving 
the  approximate  inductance  and  capacity  of  three-core  cables 
may  be  found  in  the  electrical  engineering  Handbooks  and  in 
makers'  catalogues. 

The  reactive  voltage  drop  per  mile  of  single  conductor  of  a 
three-core  cable  may  be  calculated  by  means  of  formula  (29)  of 
Article  45,  Chapter  IV,  which  is  correct  for  the  usual  frequencies, 
even  when  the  distance  (d)  between  centers  of  cores  is  as  small 
relatively  to  the  radius  (r)  of  the  conductor  as  in  three-core 
underground  cables. 

113.  Capacity  and  Charging  Current  of  Three-core  Cables. — 
Mathematical  formulas  for  calculating  the  electrostatic  capacities 
of  three-core  cables  are  complicated  and  not  very  reliable  owing 
to  certain  assumptions  having  to  be  made  which  are  not  always 
satisfied  in  a  practical  cable.     The  figures  relating  to  capacities 
of  cables,  as  furnished  by  cable  makers,  are  therefore  based  on 
test  data  obtained  from  the  finished  cable. 

The  capacity  between  two  parallel  overhead  wires,  in  micro- 
farads per  mile,  is 

1  British  patentd  No.  27858/08  and  No.  27859/08. 


218  ELECTRIC  POWER  TRANSMISSION 


. 

Cm  (between  wires)  =  (84) 

los- 

and  the  charging  current  (r.m.s.  value),  on  the  sine-wave 
assumption,  is 

Ic  =  2irfCmE  X  10-6  X  L  amperes  (85) 

where  E  is  the  (r.m.s.)  voltage  between  wires,  and  L  the  distance 
of  transmission  in  miles,  while  d  and  r  in  formula  (84)  stand  re- 
spectively for  the  distance  between  centers  of  the  conductors 
and  the  radius  of  the  conductor  cross-section. 

If  three  wires  occupy  the  corners  of  an  equilateral  triangle 
the  capacity  as  measured  between  wires  is  still  given  by  formula 
(84),  and  the  capacity  between  wire  and  neutral  will  be  just 
twice  this  value,  or 

Cm  (to  neutral)  =  ^^  (12) 


as  given  by  formula  (12)  Article  10,  Chapter  II.     The  voltage 

E  E 

across  this  imaginary  condenser  is  now  —7=  instead   of  ^  as  it 

would  be  in  the  case  of  a  single-phase  transmission,  and  the 
charging  current  per  wire  of  the  three-phase  system  is  therefore 

Ic  =  2irf^=  Cm  X  10-6  X  L         amperes          (13) 
v  3 

as  given  by  formula  (13)  of  Chapter  II.     In  this  formula,  the 
capacity  Cm  to  neutral  is  just  twice  as  great  as  the  capacity 

between  wires  as  given  by  formula  (84),  whence  it  follows  that 

o 

the  charging  current  per  wire  of  the  three-phase  system  is  —  -p. 

v  3 

times  greater  than  that  of  the  single-phase  system  with  the  same 
spacing  and  line  voltage. 

In  overhead  systems,  the  capacity  to  ground  is  generally 
negligible;  but  in  a  cable  transmission  with  or  without  lead 
sheath,  the  condenser  formed  by  the  comparatively  small  thick- 
ness of  insulation  between  conductor  and  ground  must  be 
reckoned  with.  Fig.  81  is  a  section  through  a  three-core  h.t. 
cable  with  lead  sheath.  The  capacity  of  each  of  the  imaginary 
condensers  shown  in  the  diagram  cannot  be  measured  directly; 
but  certain  measurements  can  be  made  on  the  finished  cable, 
from  which  the  necessary  data  may  be  obtained, 


TRANSMISSION  BY  UNDERGROUND  CABLES    219 

Let  Ce  stand  for  the  effective  equivalent  capacity   per   con- 
ductor to  neutral  for  one  mile  of  cable.     Then,  since  the  voltage 
•pi 

to  neutral  is  — -T=>  the  charging  current  per  conductor,  on  the  sine- 
v3 

wave  assumption,  is 

Ic  =  2irf^=  Ce  X  10-6  X  L        amperes.  (86) 

v  3 

Two  measurements  of  capacity  can  readily  be  made  on  the  fin- 
ished cable: 

(a)  between  one  core  (1)  and  the  two  remaining  cores  and  the 
lead  sheath,  all  connected  together  (2,  3,  £). 

(6)  between  any  two  cores,  as  (1)  and  (2). 


FIG.  81. — Diagrammatic  representation  of  electrostatic  capacities  in  three-core 
cable. 

In  terms  of  the  imaginary  capacities  Ca  and  Cn  of  Fig.  81,  we 
have: 

Capacity  (a)  =  C,  +  -~*-T~  =  C,  +  \  Cn  (87) 

1      ,       1  o 

C~n          2C~n 

and 

Capacity  (6)  =  i  (C,  +  C,)  (88) 

There  is  no  constant  ratio  between  wire  to  wire  capacity  and  the 
wire  to  sheath  capacity;  but,  generally  speaking,  the  former 
(6)  is  from  57  to  68  per  cent,  of  the  latter  (a).  The  most  usual 


220  ELECTRIC  POWER  TRANSMISSION 

value  is  60  per  cent.,  whence,  by  equating  (88)  with  0.6  times  (87), 
it  follows  that  C3  =  Cn. 

The  standard  test  for  electrostatic  capacity  is  between  one  core 
and  the  two  remaining  cores  grounded  to  the  sheath,  namely 
(a)  as  expressed  by  formula  (87),  and  since  the  total  capacity 
(Ce)  per  core  to  neutral  is  obviously 

Ce  =  C8  +  Cn  (89) 

it  follows  that  the  approximate  ratio  of  these  two  capacities  is, 

Effective  equivalent  capacity,  core  to  neutral  (C«) 

Measured  capacity,  one  core  to  the  remaining  cores  and  sheath  (a) 
_  (1  +  1)  X  3  _ 

3  +  2 

The  measured  capacity  (a)  for  3-core  paper-insulated  cables 
designed  for  a  working  pressure  of  10,000  volts  will  be  about 
0.4  microfarad  per  mile  for  a  3J^-o83  in.  cable,  and  0.26  for  a 
1%-058  in.  cable.  With  shaped  instead  of  circular  cores,  the 
capacity  is  slightly  greater,  being  from  1.08  to  1.1  times  the 
capacity  with  circular  cores  of  the  same  cross-section. 

Calculation  of  Capacity  of  Three-core  Cables. — Although  meas- 
ured values  of  cable  capacities  are  usually  obtainable  from 
manufacturers,  it  may  be  necessary  to  determine  approximately 
the  capacity  of  a  cable  for  a  special  purpose  before  it  has  been 
made.  It  would  seem  at  first  sight  from  Fig.  81  and  the  fact 
that  Cs  is  found  to  be  approximately  equal  to  Cn,  that  Ce  would 
be  obtained  by  merely  doubling  the  capacity  to  neutral  as  calcu- 
lated by  the  usual  transmission  line  formulas  (see  Article  52, 
Chapter  IV) .  It  should  be  observed,  however,  that  the  proximity 
of  the  sheath  will  modify  the  distribution  of  the  electrostatic  flux 
between  core  and  core,  and  the  imaginary  capacity  Cn  of  Fig.  81 
is  not  what  it  would  be  if  the  lead  sheath  were  removed  and  re- 
placed by  a  considerable  extra  thickness  of  insulation. 

It  is  found  in  practice  that  the  capacity  of  a  three-core  cable 
with  shaped  cores,  having  the  same  thickness  of  insulation 
between  core  and  core  as  between  core  and  sheath,  is  about  the 
same  as  the  capacity  of  a  single-core  cable  having  the  same 
conductor  cross-section  and  the  same  thickness  of  insulation 
between  core  and  sheath.  On  this  assumption,  we  can  use 
formula  (80)  of  Article  109  for  predetermining  the  probable 
capacity  of  a  three-core  cable. 

The  value  of  R  in  formula  (80)  is  taken  as  the  radius  of  the 


TRANSMISSION  BY  UNDERGROUND  CABLES    221 

conductor  plus  the  thickness  of  insulation  between  cores,  or 
between  core  and  sheath.  If  these  thicknesses  are  not  exactly 
the  same,  the  mean  of  the  two  thicknesses  is  taken.  If  the  con- 
ductor is  not  circular  in  cross-section,  the  dimension  r  in  the  for- 
mula is  the  radius  of  a  circular  core  of  the  same  cross-section  as 
the  actual  conductor. 

As  an  example,  let  the  diameter  of  each  core  of  a  three-core 
paper-insulated  cable  be  0.58  in.,  with  insulation  between  cores 
0.38  in.  thick,  and  between  core  and  sheath,  0.30  in.  thick. 
Assuming  the  specific  inductive  capacity  of  the  insulation  to  be 
3.3,  we  have, 

k  =  3.3 

r  =  0.29 
and 


fi  =  0.29  +-  -      =  0.63 

whence, 

„         ..    ,  .       0.0388  X  3.3      _  00     .      ,  „ 

Capacity  (a)  =  --  =0.38  microfarad  per  mile. 

log  (029) 

The  approximate  value  of  the  capacity  to  neutral,  for  shaped 
cores,  will  therefore  be, 

Ce  =  0.38  X  1.2  =  0.456 
and  for  cores  of  circular  cross-section, 

Ce  =  -r    x-  =  0.422  microfarad  per  mile. 
1.08 

114.  Example  of  Design  of  Single-phase  Concentric  E.H.T. 
Power  Cable.  —  Let  the  working  pressure'be  100  k.v.  (alternating) 
between  the  inner  and  outer  conductors.  The  further  assump- 
tion will  be  made  that  the  maximum  stress  must  not  exceed  40 
k.v.  (r.m.s.)  per  centimeter.  This  is  a  low  value,  and  50  k.v. 
would  probably  be  permissible;  but  the  lower  figure  has  been 
chosen  in  order  to  provide  a  large  factor  of  safety  and  keep  the 
dielectric  loss  (and  heating)  in  the  neighborhood  of  its  lowest 
practical  limit. 

Consider  first  the  case  of  a  cable  without  either  "voltage" 
or  "capacity"  grading.  By  formula  (83) 

E       100 
r  =  G=10=2-0m- 

If  a  solid  core  of  stranded  cable  were  used,  this  would  correspond 


222  ELECTRIC  POWER  TRANSMISSION 

to  a  cross-section  of  about  2%  sq.  in.  which  would  certainly  be  in 
excess  of  the  requirements,  and  a  hollow  core,  constructed  as 
shown  in  Fig.  80,  should  be  adopted. 
Solving  for  R  in  formula  (61),  we  have, 

100 

"  40  X  2.5 
whence 

^  =  2.718,  and  R  =  6.8  cm. 

The  dimensions  of  the  cable,  in  inches,  would  be  approximately : 
Diameter  over  outer  conductor  =  5.56  in. 
Diameter  over  lead  sheath          =  5.93  in. 
Diameter  over  armor  and  jute    =  6.6  in. 

Consider  now  the  alternative  of  intersheath-  or  voltage-grading. 

If  a  pressure  of  50  k.v.  is  maintained  on  each  side  of  a  single 
metallic  intersheath,  the  radius  r  will  now  be, 

r-  f0  =  1.25  cm. 

To  calculate  the  radius  over  the  insulation  between  the  inter- 
sheath and  the  inner  conductor,  we  have 


1        /  R  \  50 

gf  U.25/   ~  40  X  1 


u.zo/        ^u  A  i.zo 
whence 

R  =  2.718  X  1.25  =  3.4  cm. 

The  radial  thickness  of  the  lead  intersheath  would  be  about 
0.05  in.,  or  0.127  cm.,  whence  the  outside  radius  of  the  inter- 
sheath is  r'  =  3.4  +  0.127  =  3.527  cm.  Considering  this  as 
the  core  of  a  cable  with  50  k.v.  across  the  total  thickness  of  insula- 
tion, and  the  same  maximum  voltage  gradient  as  before,  we 
have  for  the  radius  over  the  insulation, 

Kn 

40^527  -  °'354 
whence 

2~±  =  1-425  and  R'  =  5.02  cm. 

The  approximate  dimensions  of  this  cable,  in  inches,  would  be 
as  follows: 

Diameter  over  outer  conductor  =  4.08  in. 
Diameter  over  lead  sheath  =  4.45  in. 

Diameter  over  armor  and  jute    =  5.12  in. 


TRANSMISSION  BY  UNDERGROUND  CABLES    223 

In  order  to  realize  the  advantage  of  this  method  of  grading 
large  high-voltage  cables,  these  figures  should  be  compared  with 
those  previously  calculated  for  the  cable  without  voltage  grading. 

It  will  be  interesting  to  calculate  the  charging  currents  per 
mile  of  this  cable.  Assuming  the  specific  inductive  capacity  of 
the  paper  insulation  to  be  k  =  3.3  as  given  on  page  213,  the  ca- 
pacity per  mile  between  core  and  intersheath,  by  formula  (80), 

is  Cm  =  -j^ 2  718    =  0.295  microfarad.     If  the  frequency  is 

25  cycles  per  second,  the  charging  current,  on  the  sine-wave 
assumption,  will  be 

Ic  =  27T  X  25  X  0.295  X  50000  X  lO"6  =  2.32  amp. 
Similarly,  between  the  intersheath  and  the  outer  conductor, 
we  have 

_  0.0388  X  3.3  _ 
Cm  ~      log  1.425 
and 

coo 

Ic  =  2.32  X  ^i  =  6.55  amp. 

ZvO 

The  difference  between  these  two  values  of  capacity  current 
must  obviously  be  carried  by  the  intersheath,  and  for  every  mile 
of  cable  the  values  of  the  charging  current  would  be  as  follows: 

In  the  core 2.32  amp. 

In  the  intersheath   .    .    .  4.23  amp. 
In  the  outer  conductor  .  6.55  amp. 

Thus,  if  the  cable  were  10  miles  long,  the  current  in  the  inter- 
sheath— if  fed  from  one  end  only — would  be  42.3  amperes,  which 
might  be  excessive.  This  is  a  point  which  must  not  be  overlooked 
in  the  design  and  installation  of  intersheath  cables. 

115.  Losses  in  Underground  Cables. — In  addition  to  the 
PR  losses  in  the  conductors,  which  can  easily  be  calculated, 
some  loss  occurs  in  the  dielectric  of  an  underground  cable.  The 
ohmic  resistance  of  the  insulation  being  very  high,  the  losses, 
when  a  high-tension  cable  is  used  on  a  continuous  current  circuit, 
are  very  small;  but  with  alternating  currents  there  is  a  further 
loss  due  to  dielectric  hysteresis,  which  is  proportional  to  the 
frequency  of  alternation  of  the  electrostatic  field. 

The  total  charging  current  in  a  cable  may  be  considered  as 
made  up  of  two  components,  one  being  the  true  capacity  current 
of  which  the  phase  is  exactly  90  degrees  in  advance  of  the  im- 
pressed e.m.f.,  the  other  being  the  "energy"  component  in 


224  ELECTRIC  POWER  TRANSMISSION 

phase  with  the  e.m.f.  This  last  component  is  relatively  small, 
being  due  to  what  is  known  as  dielectric  hysteresis  and  not  to  the 
ohmic  resistance  of  the  dielectric,  the  effect  of  which  is  usually 
negligible.  The  dielectric  loss  is  equal  to  the  product  of  the 
voltage  and  the  "energy"  or  in-phase  component  of  the  charging 
current.  Thus 

Watts  lost  =  e.m.f.  X  charging  current  X  cos^>  (91) 
where  cos  <p  stands  for  the  power  factor  of  the  cable.  This  will 
usually  be  about  0.03  for  paper-insulated  power  cables,  at 
ordinary  working  temperatures;  but,  with  high  temperatures, 
it  is  very  much  greater.  As  a  rough  indication  of  the  manner  in 
which  the  power  factor  (and  therefore  the  dielectric  loss)  in- 
creases with  temperature,  the  following  figures  may  be  useful; 
they  refer  to  paper-insulated  cables. 


Temperature  of  insulation,  degrees  C. 

Power  factor,  per  cent. 

50 

3      to  5 

60 

5.  5  to  10 

70 

9      to  15 

80 

13      to  20 

90 

19      to  30 

As  an  example  of  how  the  losses  in  a  cable  may  be  calculated, 
we  shall  use  the  data  of  the  numerical  example  in  Article  113. 
The  calculated  capacity  per  core  to  neutral  was  Ce  =  0.422 
microfarad  per  mile.  Assuming  the  voltage  between  wires 
to  be  11,000,  the  frequency  50,  and  the  distance  of  transmission 
10  miles,  the  charging  current  per  core,  by  formula  (86),  will  be, 


c 

\/3 

=  8.4  amperes, 
whence  the  total  dielectric  loss  is,  seen  to  be 


W  =  3  X          r-    X  8.4  X  0.03  X  10~3 

=  4.8  kilowatts. 

4  8 
The  apparent  power  required  is,  however,  77     =  1^0  k.v.a., 


which  is  an  indication  of  the  size  of  generator  required  to  keep 
the  full  voltage  on  the  line  when  the  receiving  end  is  disconnected 
from  the  load. 


TRANSMISSION  BY  UNDERGROUND  CABLES    225 

116.  Temperature  Rise  of  Insulated  Cables. — The  layers  of 
insulating  material  close  to  the  conductors  will  be  hotter  than 
those  near  the  surface  of  the  cable;  but  the  difference  in  tem- 
perature between  the  conductor  and  the  external  surface  is  not 
easily  predetermined.  It  will  depend  upon  the  nature  and 
thickness  of  the  insulation,  and  also  upon  the  type  of  cable,  i.e., 
whether  two-  or  three-core,  or  concentric.  The  difference  of 
temperature  between  the  core  and  the  external  sheath  of  a  fully 
loaded  power  cable  will  usually  be  between  10°  and  25° 
C.;  but  the  actual  temperature  of  the  insulation  at  the 
hottest  parts  will  be  determined  not  only  by  the  rate  at  which 
the  heat  can  be  conducted  from  the  cores  to  the  surface  through 
the  insulation,  but  also  by  the  rate  at  which  it  can  be  radiated 
or  conducted  from  the  outside  surface  of  the  cable. 

The  best  conditions  for  cooling  will  usually  occur  with  sub- 
marine cables;  but  armored  cables  buried  direct  in  certain 
kinds  of  soil  are  also  capable  of  dissipating  large  amounts  of 
energy.  When  many  cables  are  laid  close  together  in  multiple- 
way  ducts,  it  is  not  possible  to  consider  each  cable  independently 
of  the  others,  since  the  temperature  of  the  duct  will  depend  upon 
the  total  cooling  surface  and  the  total  amount  of  energy  lost  in 
all  the  cables.  The  problem  of  calculating  temperature  rise 
is  thus  seen  to  be  a  difficult  one,  and  indeed  more  data  must  be 
accumulated  before  reliable  empirical  formulas  will  be  available. 
Very  little  can  be  said  here  that  will  be  of  service  to  the  engineer 
in  determining  exactly  what  will  be  the  safe  current  for  a  given 
cable  laid  in  a  particular  manner;  but  it  is  important  to  keep 
the  temperature  of  the  insulation  within  certain  limits  which 
cannot  be  exceeded  without  injuring  the  cable  or  leading  to  a 
greatly  increased  dielectric  loss  which  aggravates  the  trouble 
and  leads  to  rapid  deterioration  of  the  insulation.  This  limit 
may  be  set  at  about  85°  C.  for  paper-insulated  cables. 

For  pressures  up  to  20,000  volts,  the  dielectric  loss  in  three- 
phase  high-grade  paper  cables  is  so  small  as  to  be  negligible; 
and  the  permissible  PR  loss  in  the  conductors  will  depend  (1) 
upon  the  rate  at  which  heat  can  be  conducted  through  the  in- 
sulation from  the  conductor  to  the  outside  surface,  and  (2) 
upon  the  facilities  afforded  for  the  cooling  of  the  outside  surface 
of  the  cable. 

It  is  possible  to  run  the  current  densities  in  three-core  20,000- 
volt  power  cables  up  to  1000  amp.  per  sq.  in.  in  cables  of 


226  ELECTRIC  POWER  TRANSMISSION 

0.25-sq.  in.  core  section,  and  even  1500  amp.  per  sq.  in.  if 
the  core  is  not  more  than  0.1  sq.  in.  cross-section.  It  is  rarely 
safe  to  allow  the  lead  sheath  to  reach  a  temperature  greater  than 
40°  C.;  but  no  hard  and  fast  rule  can  be  laid  down  in 
this  connection.  When  the  price  of  copper  is  high,  it  is  impor- 
tant to  load  cables  up  to  the  safe  limit,  and  research  work  is  being 
carried  on  with  a  view  to  furnishing  additional  information 
on  the  heating  of  underground  cables  of  different  types  and 
under  different  conditions  of  laying. 

A  simple  case,  which  admits  of  calculation  without  a  large 
amount  of  empirical  data,  is  that  of  the  single-core  or  concentric 
cable.  Thus,  if  it  is  desired  to  calculate  the  difference  in 
temperature  between  the  core  and  external  sheath  of  such 
a  cable,  the  procedure  would  be  similar  to  the  method  followed 
in  calculating  the  ohmic  resistance,  except  that  the  "heat  con- 
ductivity" of  the  dielectric  would  take  the  place  of  the  electrical 
conductivity. 

As  an  example,  and  following  the  method  outlined  in  Article 
109  when  developing  an  expression  for  the  insulation  resistance, 
consider  a  lead-covered  single-core  paper-insulated  cable  of  core 
diameter  2r  =  0.9  in.  and  diameter  over  the  insulation  of  2R 
=  1.9  in.  The  heat  conductivity  of  the  insulation  must  be  de- 
termined experimentally,  but  we  shall  assume  that  it  is  ft  =  0.002. 
This  coefficient  may  be  defined  as  the  number  of  watts  that  will  be 
conducted  through  each  square  centimeter  of  a  slab  of  insulat- 
ing material  1  cm.  thick  when  the  difference  of  temperature 
is  1°  C. 

The  cross-section  of  this  conductor  will  be  about  0.5  sq.  in., 
the  resistance  being  0.1  ohm  per  mile,  and  if  we  assume  a  current 
density  of  1300  amp.  per  sq.  in.,  the  power  loss  per  mile  will 
be  0.1  X  (650) 2  =  42,250  watts. 

Considering  a  cylinder  of  the  dielectric  of  thickness  dx  at  a 
radius  x  from  the  center  of  the  core,  the  difference  of  temperature 
between  the  two  sides  of  this  layer  of  insulation  will  be 

Wdx 

k  (2*xl) 

where  I  is  the  length  in  which  the  loss  of  W  watts  occurs  (in  this 
example  I  =  161,000  cm.).  Thus, 


W     CRdx 
2irlkjr     x 


TRANSMISSION  BY  UNDERGROUND  CABLES    227 


42250  .        /1.9 

=  lf\or      I 

27T  X  161000  X  0.002    [*  \0.9 
=  15.5°  C. 

If  the  cable  were  suspended  in  air,  the  difference  of  temperature 
between  the  lead  sheath  and  the  surrounding  air  could  be  calcu- 
lated by  assuming  about  0.0012  watt  to  be  radiated  from  each 
square  centimeter  of  surface  per  degree  Centigrade  difference  of 
temperature.  The  outside  diameter  of  this  cable — if  there  is  no 
jute  or  steel  armoring  over  the  lead — will  be  about  2.15  in.,  and 
the  rise  in  temperature  of  the  lead  sheath  will  therefore  be 


42250 


X  2.15  X  2.54 X  161000  X  0.0012 


=  12.7  degrees. 


70 
60 

1- 

r 

g  30 
H10 

f?f,en 

u 

per  PI 

ase 

C 

urve  No.  2 

^ 

Curve 

-Norl- 

/ 

V250C 

peri- 

lase 

^ 

^ 

/ 

V  • 

/ 

J 

Time 


10 


H  Hoars 


FIG.  82. — Temperature  rise  of  33  kv.  0.25  sq.  in.  three-phase,  paper  insu- 
lated, lead-covered  cables  in  iron  pipes  laid  12  in.  apart  3  ft.  below  ground 
surface. 

The  total  temperature  rise  of  the  copper,  on  this  basis,  is 
therefore  15.5  +  12.7  =  28.2°  C.;  but  since  the  cable  is  not 
likely  to  be  suspended  in  air,  correction  coefficients  derived 
experimentally  would  have  to  be  applied  in  order  to  determine 
the  probable  temperature  rise  under  practical  conditions.  A 
further  correction  would  have  to  be  made  if  the  cable  is  provided 
with  an  outer  covering  of  jute. 

Since  the  final  temperature  rise  of  a  cable  under  given  condi- 
tions will  be  very  nearly  proportional  to  the  square  of  the  current, 
we  may  write 

P  =  KT 


228  ELECTRIC  POWER  TRANSMISSION 

and  if  the  temperature  rise  T\  is  known  for  a  given  current  /i, 
the  value  of  the  constant  K  can  be  determined,  and  the  tempera- 
ture rise  with  any  other  current  72  will  be  approximately 

T  2 

Tz  =  -R 

The  curves  of  Fig.  82  show  not  only  the  final  temperature  rise 
attained  by  three-core  paper-insulated  power  cables  drawn  in 
iron  pipes  underground,  but  also  the  time  required  to  bring  about 
any  particular  rise  in  temperature.  The  vertical  scale  indicates 
degrees  Fahrenheit  above  an  initial  temperature  of  57  degrees, 
while  the  horizontal  scale  gives  the  hours  during  which  the  cur- 
rent has  been  on  the  cable.  The  cross-section  of  each  core  of  this 
cable  is  0.25  sq.  in.,  and  the  tests  were  made  with  current  den- 
sities of  1000  and  1200  amperes  per  square  inch;  the  frequency 
of  the  supply  being  50.  The  value  of  the  constant  K  in  the  ex- 
pression 72  =  KT  as  calculated  for  /  =  250  amperes,  is  1330, 
whence  the  calculated  temperature  rise  for  300  amperes  is 
67.6°  F.  which  is  somewhat  higher  than  the  observed  rise  of 
62  degrees,  as  one  would  expect  it  to  be. 

No  appreciable  rise  of  temperature  was  observed  with  the  full 
working  voltage  on  the  cable  without  load.  In  other  words, 
the  dielectric  losses  were  not  of  such  magnitude  as  to  add  appre- 
ciably to  the  heating  caused  by  the  I2R  losses  in  the  conductors. 

117.  Reliability  of  Cable  Systems.  Joints;  Electrolysis.— 
Apart  from  mechanical  injury,  which  must  be  guarded  against 
by  giving  attention  to  the  method  of  laying  and  to  the  handling 
of  the  cables  during  their  installation,  trouble  can  usually  be 
traced  (1)  to  poor  joints,  (2)  to  general  or  local  overheating,  and 
(3)  to  electrolysis. 

Joints.— Even  when  the  trouble  is  caused  by  overheating 
rather  than  initial  weakness  of  insulation,  this  frequently  occurs 
in  the  neighborhood  of  poorly  made  joints.  A  considerable 
amount  of  skill  is  necessary  in  making  satisfactory  joints  on  h.t. 
underground  cables,  and  the  difficulty  experienced  in  obtaining 
skilled  and  reliable  workmen  has  led  to  the  development  of  de- 
signs employing  special  insulating  spacers  and  accurately  made 
metal  sleeves  or  bridge  pieces,  which  depend  less  upon  the  skill 
and  experience  of  the  jointer  than  the  older  methods  involving 
paper  or  tape  wrappings. 

Careful  bonding  of  the  lead  sheath  is  also  a  matter  of  consider- 


TRANSMISSION  BY  UNDERGROUND  CABLES    229 

able  importance  if  electrolytic  troubles  are  to  be  avoided,  and 
this  is  provided  for  in  the  designs  illustrated  by  Figs.  83  and 
84. J  The  former  shows  a  straight  joint  on  a  three-phase  cable 
suitable  for  about  20,000  volts ;  paper  sleeves  are  slipped  over  the 
joints,  and  rubber  rings  keep  these  the  proper  distance  apart. 
A  high  grade  of  insulating  compound  is  poured  inside  the  lead 


Clamp. 


Compound  Filled. 


ton* 


Bitumen  Cord 


inclosed  in  Cast  Iron  Caso, 


mg  Sleeve. 


0. 1  Box'  ^^^^         C  L.  Sle.v« 

FIG.  83. — Joint  in  high  tension  (20  kv.)  cables  laid  direct  in  ground. 

sleeve  forming  the  bond  between  the  lead  sheathings,  while 
bitumen  may  be  used  in  the  space  between  the  lead  and  the  outer 
case  of  cast  iron.  It  will  be  noted  that  all  sharp  corners  are 
avoided;  the  castings  being  designed  with  curves  of  large  radius 
to  obtain  the  proper  distribution  of  the  dielectric  flux. 

1  Designs  of  Messrs.  W.  T.  Glover  and  Co. 


230 


ELECTRIC  POWER  TRANSMISSION 


TRANSMISSION  BY  UNDERGROUND  CABLES    231 

Fig.  84  shows  a  somewhat  similar  joint  box,  for  33,000-volt 
three-core  cables.  The  special  jointing  ferrules  are  designed 
with  curved  surfaces  so  shaped  as  to  avoid  concentration  of 
stress,  and  thoroughly  vitrified  unglazed  porcelain  spacers  are 
used  between  the  cables  in  order  to  eliminate  wrappings  of  tape 
or  paper.  The  cast  lead  sleeve  forming  the  bond  between  the 
lead  sheathings  is  made  in  three  parts  to  facilitate  assembly. 

Overheating. — Apart  from  the  damage  done  to  the  insulation 
by  Very  high  temperatures,  it  is  generally  true  that  the  dielectric 
loss  increases  with  the  temperature,  and  this  is  especially  notice- 
able with  paper-insulated  cables.  It  is  nevertheless  important 
to  operate  underground  cables  at  a  reasonably  high  current 
density,  otherwise  the  interest  on  capital  expenditure  will  be 
greatly  in  excess  of  the  annual  cost  of  the  losses,  and  the  system 
will  be  economically  unsatisfactory.  The  engineer  is  therefore 
faced  with  a  problem  of  considerable  magnitude  if  underground 
transmission  is  to  be  adopted  on  a  larger  scale  than  at  present. 
General  or  local  overheating  of  cables  is  perhaps  the  chief  cause 
of  service  interruptions  on  underground  systems,  yet  a  liberal 
cross-section  of  conductor  can  hardly  be  considered  as  the  proper 
solution  of  the  problem.  Attention  to  all  the  causes  that  may 
lead  to  local  overheating  is  very  important;  but  the  matter  is  not 
one  that  lends  itself  to  lengthy  discussion  in  these  pages. 

Since  submarine  power  cables  will  transmit  much  greater 
amounts  of  energy  for  a  given  size  than  underground  cables, 
the  possibility  of  flooding  the  ducts  with  water,  as  suggested 
by  Mr.  Harper1  deserves  consideration. 

Electrolysis. — Chemical  action,  unassisted  by  electric  currents, 
very  rarely  injures  the  lead  sheath  of  underground  cables;  but 
electrolytic  corrosion  of  the  lead  covering — resulting  in  per- 
forations and  damage  to  the  insulation  by  the  admission  of 
moisture — is  not  uncommon.  An  underground  cable  laid  within 
a  few  feet  of  an  electric  railway  or  car  line,  is  liable  to  electrolysis; 
but  in  countries,  such  as  England,  where  rules  governing  the 
maximum  permissible  potential  difference  between  current- 
carrying  rails  and  ground  are  rigidly  enforced,  trouble  due  to  this 
cause  is  very  rare.  Perfect  bonding  of  the  lead  sheath  and 
also  of  the  steel  armoring  will  effectively  prevent  trouble  which 
otherwise  might  be  experienced. 

^'Problems  of  Operation  and  Maintenance  of  Underground  Cables," 
by  J.  L.  Harper,  Trans.  A.  I.  E.  E.,  p.  417,  vol.  xxxvi,  1917. 


232  ELECTRIC  POWER  TRANSMISSION 

Leakage  of  current  from  the  cables  themselves,  due  to  careless 
work  and  inefficient  sealing  at  points  where  connections  are 
made  is  another  cause  of  electrolytic  corrosion  in  underground 
distributing  systems. 

Electrolysis  of  lead  and  iron  buried  in  the  ground  is  usually 
caused  by  stray  currents  from  electrically  operated  railroads  or 
street-car  lines  which  use  continuous  currents.  The  effects  of 
alternating  currents  of  frequencies  between  15  and  60  cycles  is 
practically  negligible,  being  least  with  the  highest  frequency. 

The  actual  amount  of  metal  carried  away  from  the  surface 
of  the  anode  (usually  lead  or  iron)  will  depend  not  only  upon  the 
surface  exposed  and  the  density  of  the  stray  currents  which 
cause  the  corrosion;  but  also  on  the  nature  of  the  soil  and 
whether  it  is  usually  moist  or  dry. 

The  reader  who  desires  to  investigate  more  fully  the  subject 
of  electrolytic  corrosion  is  referred  to  the  important  contribution 
by  Messrs.  McCollum  and  Ahlborn.1  Their  paper  includes 
references  to  the  work  of  previous  investigators,  and  brings  out 
very  clearly  the  importance  of  reversing  the  direction  of  the 
currents  through  the  ground,  even  if  the  period  of  such  reversal 
is  measurable  in  hours  or  days.  The  shifting  of  loads  on  electric 
railway  systems  in  cities  usually  produces  large  areas,  called 
neutral  zones,  where  the  polarity  of  underground  pipes  reverses 
at  more  or  less  frequent  intervals.  Even  when  such  intervals 
of  reversal  range  from  several  minutes  to  one  or  two  hours, 
the  corrosion  due  to  electrolysis  is  found  to  be  less  than  would 
be  expected  if  attributed  to  the  average  amount  of  current  dis- 
charged from  the  pipes  into  the  earth.  It  would  seem  as  if 
the  reversal  of  current  actually  causes  metal  to  be  redeposited 
on  the  corroded  portions,  even  after  the  pipes  have  acted  as  anodes 
during  a  considerable  lapse  of  time.  The  redeposited  metal 
will  probably  have  little  effect  in  strengthening  the  pipe 
mechanically;  but  it  will  serve  as  an  anode  surface  during  the 
succeeding  period  of  current  discharge,  and  thus  protect  the 
uncorroded  metal  beneath,  which  otherwise  would  have  been  at- 
tacked. This  action,  due  to  comparatively  slow  reversals  of  cur- 
rent, will  be  interfered  with  by  circulation  of  the  electrolyte,  and 
the  action  of  air  (oxygen  and  carbon  dioxide)  on  the  corroded  metal. 

x"  Influence  of  Frequency  of  Alternating  or  Infrequently  Reversed 
Current  on  Electrolytic  Corrosion,"  by  Burton  McCollum  and  G.  H.  Ahl- 
born. Technological  Paper  No.  72  of  the  Bureau  of  Standards,  Washington. 


CHAPTER  VIII 

TRANSMISSION  OF  ENERGY  BY  CONTINUOUS 
CURRENTS 

118.  General  Description  of  the  Thury  System. — In  the  Thury 
system  of  electric  power  transmission  by  continuous  currents, 
the  current  is  constant  in  value  and  the  pressure  is  made  to  vary 
with  the  load.  All  the  generators  and  all  the  motors  are  con- 
nected in  series  on  the  one  wire,  which  may  be  in  the  form  of 
a  closed  loop  serving  a  wide  area,  or  it  may  consist  merely  of  the 


Mwhlue  Voltmeters 
Short  Circuiting 


Motor      TwoMotori       Two  Motors 


FIG.  85. — Diagram  of  connections — Thury  series  system. 

outgoing  and  returning  wires  between  a  power  station  containing 
all  the  generators,  and  one  or  more  substations,  with  motors,  at 
the  end  of  a  direct  transmission.  The  diagram,  Fig.  85,  shows  a 
typical  arrangement  of  machines  for  a  small  installation  on  the 
Thury  system.  In  this  example  there  are  four  generators  at  one 
end  of  the  line,  and  seven  motors  of  various  capacities  at  the  other 
end  of  the  line,  all  the  machines  being  connected  in  series  and  pro- 

233 


234  ELECTRIC  POWER  TRANSMISSION 

'  vided  with  short-circuiting  switches.  The  connections  are, 
however,  so  simple  that  the  diagram  is  self-explanatory.  The 
voltage  at  the  terminals  of  any  one  dynamo  is  limited,  as  the 
necessity  for  a  commutator  renders  it  impossible  to  wind  a  con- 
tinuous-current machine  for  so  high  a  pressure  as  may  be  ob- 
tained from  alternating-current  machines.  The  limiting  pressure 
per  commutator  on  the  existing  Thury  systems  is  5000  volts, 
this  being  on  the  machines  of  the  Metropolitan  Electric  Supply 
Company  of  London:  the  lowest  is  1300  on  a  small  two-generator 
plant  in  Russia. 

In  order  to  obtain  the  high  pressures  required  for  economical 
transmission  over  long  distances,  it  is  necessary  to  connect  many 
generator  units  in  series;  the  difficulty  of  insulation  between 
machines  and  ground  being  overcome  by  mounting  the  dynamos 
and  motors  on  insulators  and  providing  an  insulated  coupling 
between  the  electric  generators  and  the  prime  movers.  An  in- 
sulating floor  is  also  provided. 

When  a  machine,  whether  generator  or  motor,  is  not  in  use, 
it  is  short-circuited  through  a  switch  provided  for  this  purpose. 
As  the  motor  load  varies,  generators  are  switched  in  or  out  of 
circuit,  thus  varying  the  total  voltage.  When  it  is  required 
to  switch  in  an  additional  generator,  the  machine  is  brought  up 
to  speed  until  it  gives  the  proper  line  current  before  the  short- 
circuiting  switch  is  opened  to  throw  the  machine  in  series  with 
the  line.  To  start  up  a  motor,  the  short-circuiting  switch  is 
opened  when  the  brushes  are  in  the  position  of  zero  torque. 
The  brush  rocker  is  then  gradually  moved  round,  and  the  motor, 
starting  from  rest,  increases  in  speed  until  the  brushes  are  in  the 
required  position;  the  actual  speed,  for  any  particular  position 
of  the  brushes,  being  dependent  upon  (he  load. 

The  motors  may  be  distributed  anywhere  along  the  line, 
either  on  the  premises  of  private  users  of  power,  where  they  may 
be  directly  coupled  to  the  machinery  to  be  driven,  or  in  sub- 
stations, coupled  to  constant  pressure  electric  generators  giving 
a  secondary  supply  for  lighting  and  power  purposes.  In  most 
of  the  Thury  undertakings  in  Europe,  this  secondary  supply  is 
three-phase  alternating  current. 

A  series-wound  dynamo  machine,  with  a  current  of  constant 
value  passing  through  field-magnet  and  armature  windings,  is 
essentially  a  constant  torque  machine.  In  the  case  of  a  motor, 
if  the  load  is  decreased,  the  motor  will  increase  in  speed  and  tend 


TRANSMISSION  BY  CONTINUOUS  CURRENTS  235 

to  "run  away"; 'with  increase  of  load,  the  motor  will  slow  down, 
and  in  time  come  to  a  stand-still.  In  regard  to  the  generators, 
the  ideal  prime  mover  is  one  which  will  give  a  constant  torque, 
such  as  a  steam  engine  with  fixed  cut-off  and  constant  steam 
pressure;  and  a  single  generator  so  driven  would  be  practically 
self-regulating,  and  maintain  constant  current  regardless  of  load, 
as  the  speed — and  therefore  the  pressure  at  terminals — would 
adjust  itself  to  suit  the  motor  load.  The  generators  are,  however, 
usually  driven  by  prime  movers  which  are  far  from  fulfilling  the 
ideal  conditions.  Most  of  the  Thury  stations  are  driven  by 
water  turbines,  which  are  most  efficient  as  constant  speed  ma- 
chines; while  the  maximum  torque  at  low  speeds  is  generally 
about  twice  the  torque  under  conditions  of  highest  efficiency  at 
normal  speed. 

Just  as  various  devices  are  provided,  when  working  on  the 
parallel  system,  to  maintain  constant  pressure  of  supply,  so  in 
the  series  system,  it  is  necessary  to  provide  regulating  devices  to 
maintain  a  constant  current.  Regulators  controlled  by  the  main 
current,  or  by  a  definite  fraction  of  the  main  current,  passing 
through  a  solenoid,  can  be  made  to  act  on  mechanism  designed 
to  vary  the  speed  of  the  prime  movers.  This  method  is  quite 
practicable,  but,  where  the  type  of  engine — such  as  a  water 
wheel  under  constant  head — is  not  suited  to  variable  speed  run- 
ning, the  machines  may  be  run  at  constant  speed,  and  the  auto- 
matic device  made  to  alter  the  magnetic  field  cut  by  the  armature 
conductors,  this  being  the  only  alternative  means  of  varying 
the  voltage  generated.  The  alteration  of  the  effective  magnetic 
flux  may  be  effected: 

(1)  By  shunting  a  portion  of  the  main  current  so  that  it  shall 
not  all  pass  through  the  field  winding,  and 

(2)  By  shifting  the  position  of  the  brushes  on  the  commutator. 
A  combination  of  both  methods  appears  to  give  satisfactory 

results.  The  method  (1)  alone  is  liable  to  lead  to  sparking 
troubles  because  of  the  relatively  greater  armature  reaction  due 
to  the  weakening  of  the  field;  and,  in  practice,  it  is  found  inad- 
visable to  shunt  more  than  one-third  of  the  total  current.  It  is, 
of  course,  understood  that  the  large  variations  of  voltage  are 
obtained  by  connecting  more  or  fewer  generators  in  series  on  the 
line. 

The  motors,  whether  connected  directly  to  the  machinery  of 
mills  or  factories,  or  used  for  driving  sub-generators  of  the  con- 


236  ELECTRIC  POWER  TRANSMISSION 

slant  pressure  type,  are  usually  required  to  run  ac  constant  speed. 
Their  regulation  is  effected  by  a  small  centrifugal  governor  which 
rocks  the  brushes  by  acting  on  intermediate  mechanism  driven 
by  the  motor  itself.  The  reversal  of  a  motor  is  most  simply 
effected  by  shifting  the  brushes  round,  through  the  no-voltage 
position,  until  the  current  reverses  in  the  armature  coils. 

A  short-circuit  on  a  motor  merely  removes  that  portion  of 
the  total  load  from  the  system,  and  the  regulators  on  the  gener- 
ators will  readjust  the  pressure  accordingly.  If  a  short-circuit 
occurs  on  a  generator,  the  prime  mover  may  be  protected  from 
the  shock  by  a  slipping  coupling,  which  is  commonly  provided. 
If,  owing  to  the  failure  of  a  prime  mover,  a  generator  tends  to 
reverse  and  be  driven  as  a  motor,  it  may  be  short-circuited  by  a 
switch  that  can  very  easily  be  made  to  operate  automatically  on 
reversal  of  current. 

119.  Straight  Long-distance  Transmission  by  Continuous  Cur- 
rents.— Although  high-pressure  direct  current  may  be  used 
on  the  loop  system  with  any  number  of  motors  or  motor  substa- 
tions distributed  along  the  line — and,  if  desired,  with  any  number 
of  generating  stations  at  suitable  points  on  the  loop — it  will 
generally  be  found  that  a  parallel  constant-pressure  system  is 
preferable  for  covering  a  large  industrial  area,  the  simple  reason 
being  that,  with  the  series  system  having  a  load  more  or  less 
uniformly  distributed  along  the  loop,  the  system  is  a  high-ten- 
sion transmission  at  the  start  only,  since  the  required  voltage 
decreases  with  the  distance  from  the  generating  plant.  It  is  true 
that  the  cost  of  the  insulation  may  therefore  be  less  than  for  a 
system  on  which  the  pressure  is  high  throughout,  but  that  can  be 
said  of  any  low-tension  system.  The  point  is  that,  in  the  case  of 
the  series  loop  serving  a  wide  district,  with  power  taken  off  at 
intervals  along  the  line,  the  average  pressure  at  which  power  is 
supplied  to  the  motors  or  substations  is  only  about  half  that 
which  is  supplied  to  the  line  where  it  leaves  the  power  station. 
It  must  not  be  concluded  that  the  Thury  system  is  not  well 
adapted  to  supplying  several  motor  substations.  It  is  an  easy 
matter,  as  previously  mentioned,  to  connect  any  number  of 
motors  in  series  on  the  line,  but  in  order  to  get  the  full  benefit  of 
the  series  system  these  substations  should  all  serve  a  compara- 
tively small  district  at  the  distant  end  of  the  transmission  line. 

Apart  from  these  considerations,  and  notwithstanding  the 
advantages  of  a  long-distance  straight  transmission,  the  Thury 


TRA  NSMISSION  B  Y  CON  TIN  UO  US  C  URRENTS  237 

system  would  appear  to  be  admirably  adapted  to  serve  as  a  link 
between  otherwise  isolated  power  plants  in  an  industrial  or  thickly 
populated  district  of  considerable  •  area.  By  means  of  this 
system  there  is  not  the  slightest  difficulty  in  putting  in  series  a 
number  of  generating  stations  on  the  one  power  line,  and  stations 
supplying  alternating  current  of  various  voltages  and  frequencies 
can  thus  be  linked  together  with  the  greatest  ease  and  simplicity. 
120.  Insulation  of  Line  when  Carrying  Continuous  Currents. — 
The  question  of  sparking  distances  and  the  behavior  of  insulating 
materials  when  subjected  to  continuous-current  pressures  of 
high  values  is  of  the  greatest  importance  when  considering  the 
relative  values  of  the  Thury  system  and  the  more  common  three- 
phase  high-tension  transmission.  On  the  assumption  of  the 
theoretical  sine  wave,  the  maximum  instantaneous  value  of  an 
alternating  e.m.f.  is  \/2  times  the  root-mean-square  value,  and 
comparisons  between  alternating-current  and  direct-current 
transmissions  are  usually  made  on  this  basis,  which  makes  the 
allowable  continuous-current  pressure  to  ground  or  between 
wires,  for  the  same  insulation  and  spacing,  V2  times  the  working 
pressure  of  an  alternating-current  system.  The  ratio  should, 
however,  be  based  on  experimental  data,  and,  with  a  view  to 
obtaining  definite  and  conclusive  information  on  this  point,  Mr. 
Thury  conducted  some  years  ago  a  very  complete  set  of  com- 
parative tests  with  high  voltages,  both  continuous  and  alter- 
nating. The  results  of  these  tests  are  probably  more  favorable  to 
the  alternating-current  systems  than  would  have  been  the  case 
ha.d  they  been  conducted  on  existing  high-pressure  power-trans- 
mission systems,  because  the  experimental  alternator  used  in 
the  tests  gave  a  rather  flat-topped  e.m.f.  wave  without  any 
irregularities.  The  tests  conducted  to  determine  the  comparative 
pressures  at  which  various  insulating  materials  would  be  punc- 
tured all  tend  to  show  that,  with  continuous  currents,  something 
more  than  twice  the  alternating  pressure  is  required  to  puncture 
the  insulation;  and,  in  regard  to  sparking  distances,  the  direct- 
current  voltage  necessary  to  spark  over  a  given  distance  is,  on 
the  average,  double  the  alternating-current  voltage.  In  fact, 
this  very  complete  series  of  tests  seems  to  indicate  that  any 
existing  transmission  line  designed  for  a  definite  maximum 
working  pressure  with  alternating  currents  is  capable  of  being 
used  to  transmit  continuous  currents  at  twice  this  pressure.  It 
is  also  interesting  to  note  that  insulators  which  become  hot  when 


238  ELECTRIC  POWER  TRANSMISSION 

subjected  to  high  alternating-current  voltages  remain  cool  when 
tested  with  continuous  currents.  In  fact,  the  leakage  losses  on 
the  Thury  transmissions  are  small.  The  total  leakage  loss  over 
about  3000  insulators  on  the  St.  Maurice-Lausanne  transmission 
(a  distance  of  35  miles),  even  in  damp  weather,  is  something  of 
the  order  of  900  watts. 

It  is  usual  to  employ  two  insulated  wires  for  direct-current 
high-pressure  transmission,  but  under  certain  conditions  it 
might  be  quite  satisfactory  to  use  the  earth  as  the  return  con- 
ductor. The  arrangement  with  two  wires  and  the  entire  electric 
circuit  insulated  from  earth  is  usual  for  pressures  up  to  25,000 
volts.  It  has  the  advantage  over  any  grounded  system  that  any 
point  on  the  circuit  may  become  grounded  without  causing  a 
stoppage,  and  repairs  can  readily  be  carried  out  by  temporarily 
grounding  two  more  points,  one  on  each  side  of  the  fault.  The 
facility  and  safety  with  which  repairs  on  the  high-tension  system 
can  be  carried  out  by  grounding  the  point  where  the  work  is 
being  done  is  another  advantage  of  this  arrangement. 

If  a  ground  connection  is  made  at  both  ends  of  the  two-wire 
transmission,  the  ground  wire  being  so  situated  as  to  balance 
the  load  as  well  as  possible,  an  arrangement  equivalent  to  the 
ordinary  three-wire  system  is  obtained.  The  pressure  between 
wires  may  then  safely  be  doubled  because  the  potential  difference 
between  any  one  wire  and  earth  can  never  exceed  half  the  maxi- 
mum pressure  of  transmission.  On  the  other  hand,  some  of  the 
advantages  of  the  non-grounded  system  are  lost. 

A  direct-current  transmission  to  any  economic  distance  fyy 
means  of  a  single  wire,  using 'the  earth  as  the  return  conductor, 
is  by  no  means  an  impossible  scheme.  The  ground  resistance 
is  practically  zero,  the  loss  of  pressure  being  almost  entirely  in 
the  immediate  neighborhood  of  the  grounding  plates.  Tests 
made  on  the  St.  Maurice-Lausanne  line  (35  miles)  gave  a  total 
ground  resistance  of  0.5  ohm.  Continuous  currents  of  the  order 
of  100  amp.  returning  through  the  earth  do  not  appear  to  be 
objectionable  in  any  way.  By  taking  the  ground  connections 
to  a  considerable  depth  below  the  surface,  the  current  density  at 
ground  level  would  everywhere  be  so  small  that  interference  with 
opposing  interests  would  hardly  be  possible. 

121.  Relative  Cost  of  Conductors:  Continuous  Current  and 
Three-phase  Transmissions. — In  order  to  study  the  relative 
costs  of  conductor  material  required  for  the  series  direct-current 


TRANSMISSION  BY  CONTINUOUS  CURRENTS  239 


system  and  the  more  common  three-phase  alternating-current 
transmission,  a  basis  of  comparison  is  necessary,  and  the  following 
assumptions  will  be  made: 

(A)  Same  distance  of  transmission;  no  tapping  of  current  at 
intermediate  points. 

(B)  Same  total  amount  of  power  transmitted. 

(C)  Same  power  loss  in  conductors  (losses  due  to  leakage  and 
capacity  of  lines  are  neglected). 

(D)  Same  insulation  used  on  both  systems. 


o 


73 


FIG.  86.  FIG.  87. 

FIGS.  86  and  87. — Comparison  of  voltages  on  direct-current  and  three-phase 
systems. 

This  last  condition  is  practically  equivalent  to  stating  that  the 
maximum  value  of  the  voltage  shall  be  the  same.  It  is  proposed 
to  consider  the  following  four  conditions: 

(a)  Same  maximum  pressure  above  ground;  the  direct-current 
voltage  being  \/2  times  the  alternating-current  voltage   (sine 
wave  assumed). 

Ratio  ~  =  -4=  (92) 

E       ^/2 

where  E  and  Ea  stand  respectively  for  the  continuous  and 
alternating  voltages  to  ground.     (See  Figs.  86  and  87.) 

(b)  Same  as  (a);  but  direct-current  voltage  double  the  alter- 
nating voltage. 

"r  =  £  (93) 


240  ELECTRIC  POWER  TRANSMISSION 

(c)  Same  pressure  between  wires;  the  allowable  direct-current 
pressure  being  \/2  times  the  alternating-current  pressure. 

a      V2 

= 


(d)  Same  as  (c)  ;  but  direct-current  pressure  double  the  alter- 
nating-current pressure. 

1  .    Ea       1  (  n 

=  -,  or  ratio  ¥=^= 

To  satisfy  the  condition  of  equal  total  power,  the  equation  is, 

2E  X  I  =  ZEJa  cos  d  (96) 
and  for  equal  line  losses, 

2PR  =  3Ia*Ra  (97) 

where  I  is  the  current  per  conductor  in  the  direct-current  trans- 
mission, and  R  the  resistance  per  mile  of  single  conductor;  while 
I  a  and  Ra  are  the  corresponding  quantities  for  the  three-phase 
transmission. 

In  either  system  the  total  weight  (and  cost)  of  the  conductors 

number  of  conductors 

is  proportional  to  —  =-r-       —  *  -  r  --  j  —  —  which  gives  the 
resistance  of  each  conductor 

relation, 

Cost  of  conductors,  direct-current  system  _  2Ra 
Cost  of  conductors,  three-phase  system     '*  3R 

but  Ra  can  be  expressed  in  terms  of  R  thus: 
By  (97) 

ff         2/«fi 

=  ~&r* 

and  by  (96) 

ZEglg     COS     8 

2E 
or 

_  9ffa2/a2  cos2  6 
4E* 

which,  when  put  for  Z2  in  formula  (99),  gives, 

3ff02  cos2  0  X  R  , 

a  =         —  2E*  — 

Thus  the  equation    (98)  becomes, 

Cost  of  conductors,  direct-current  system  _  Ea2       2 
Cost  of  conductors,  three-phase  system    ~  E2 


TRANSMISSION  BY  CONTINUOUS  CURRENTS  241 

Assuming  the  very  common  value  of  0.8  for  the  power-factor 
of  the  three-phase  system,  the  numerical  ratio  for  the  four  con- 
ditions previously  stated  would  be: 

(a)  For  same  maximum  pressure  to  ground,  with  sine  wave 

assumption, 

Direct-current  cost       _  cos2  0  _ 
Alternating-current  cost  ~      2 

(b)  Same  as  (a),  but  allowable  direct-current  pressure  assumed 

to  be  double  the  alternating-current  pressure, 

Direct-rcurrent  cost       _  cos2  0 
Alternating-current  cost          4 

(c)  For  same  maximum  pressure  between  wires,   with  sine 

wave  assumed, 

Direct-current  cost  2 

=  =  cos2  0  =  0.426 


Alternating-current  cost       3 
(d)  Same  as  (c),  but  allowable  direct-current  pressure  assumed 
double  the  alternating-current  pressure, 

Direct-current  cost  1 

-  cos2  6  =  0.213 


Alternating-current  cost  3 
The  transmission  line,  apart  from  the  cost  of  conductors,  would 
be  cheaper  for  the  direct-current  than  for  the  three-phase  scheme 
because  there  are  fewer  insulators  required  and  only  two  instead 
of  three  conductors  to  string;  and  if  a  grounded  guard  wire  is 
erected  above  the  conductors,  it  is  more  convenient  to  arrange 
this  over  the  two  direct-current  conductors  than  over  the  three 
alternating-current  wires,  and  it  would  not  necessitate  the  same 
total  height  of  tower.  The  important  saving  is,  however,  in  the 
conductors  themselves.  Taking  the  figure  most  favorable  to  the 
direct-current  scheme  (b),  the  alternating-current  conductors  to 
transmit  the  same  power  with  the  same  loss  would  cost  six  and 
a  quarter  times  as  much  as  if  direct-current  transmission  were 
used,  and  even  under  the  assumption  (c),  most  favorable  to  the 
three-phase  scheme,  the  cost  would  still  be  2.35  times  the  cost 
of  the  direct-current  conductors.  For  the  purpose  of  getting 
out  preliminary  estimates,  it  is  certainly  safe  to  assume  that,  if 
the  power  factor  of  the  three-phase  load  may  be  taken  as  0.8  the 
cost  of  conductors  on  a  long-distance  direct-current  transmission 
would  be  only  one-quarter  of  the  cost  of  conductors  with  the 
alternating-current  scheme  on  the  assumption  of  equal  IZR 
losses. 


242  ELECTRIC  POWER  TRANSMISSION 

If  a  ground  return  were  used  on  a  straight  long-distance  trans- 
mission— a  perfectly  feasible  arrangement — the  cost  of  copper 
for  the  same  total  I2R  loss  would  be  only  one  quarter  of  the 
cost  of  a  two-wire  transmission,  since  the  loss  in  the  ground 
return  would  be  negligible.  The  cost  of  copper  would  then  be 
only  about  one-sixteenth  of  the  cost  of  copper  on  the  equivalent 
three-phase  transmission,  a  point  which  suggests  that  the  Thury 
system  is  worthy  of  more  serious  consideration  than  it  has 
so  far  received  outside  of  Europe.  It  is  not  suggested  that  a 
comparison  on  the  basis  of  equal  line  losses  is  necessarily  correct 
or  justifiable  on  economic  grounds;  but  this  does  not  render  the 
above  comparisons  less  interesting  or  valuable. 

122.  Concluding  Remarks  on  Direct-Current  Transmission. — 
As  an  indication  of  what  has  been  done  in  Europe  since  the  intro- 
duction of  the  Thury  system  a  quarter  of  a  century  ago,  it  may  be 
stated  that  there  are  at  present  about  16  separate  transmissions  in 
operation,  in  Switzerland,  Italy,  France,  Hungary,  Spain,  Russia 
and  England.  The  shortest  length  of  loop  is  12.4  miles  (Batoum, 
Russia),  with  a  line  pressure  of  2600  volts.  The  longest  is  248 
miles  (124  miles  straight  transmission),  this  being  the  Moutiers- 
Lyons  line  at  a  maximum  pressure  of  100,000  volts,  the  current 
being  150  amperes. 

In  England,  the  direct-current  series  system  has  been  adopted 
by  the  Metropolitan  Electric  Supply  Co.  of  London  on  their 
Western  section.  The  plant  has  been  in  operation  since  March 
of  1911  and  given  entire  satisfaction.  The  current  is  about  100 
amp.,  but  can  be  varied  from  70  to  120  amp.  without  causing 
trouble  through  sparking  on  the  commutators.  The  com- 
mutators measure  5  ft.  in  diameter  and  6  %  in.  in  length.  They 
have  1439  segments  and  run  sparklessly  at  5000  volts  between 
brushes.  The  machines  have  six  poles  and  only  two  sets  of 
brushes. 

An  interesting  account  of  the  Thury  system,  by  Mr.  William 
Baum,  with  brief  descriptions  of  the  important  European 
plants,  will  be  found  on  page  1026  of  the  General  Electric  Review 
of  Nov.,  1915. 

As  an  example  of  what  might  be  done  at  the  present  time  in 
the  way  of  direct-current  transmission  on  a  large  scale,  it  is 
clear  that  no  difficulty  need  be  experienced  in  building  dynamos 
of  a  large  size  with  5000  volts  on  one  commutator.  Assuming  a 
current  of  300  amp.,  which  would  probably  be  transmitted  by 


TRANSMISSION  BY  CONTINUOUS  CURRENTS  243 

two  conductors  connected  in  parallel,  the  output  of  each  machine 
would  be  1500  k.w.  and  two  of  these  might  be  coupled  to  one 
prime  mover.  With  two  commutators  per  machine,  the  output 
would  be  3000  k.w.,  and  with  four  commutators,  6000  k.w.  per 
unit.  Six  machines  in  series,  each  with  four  commutators,  would 
have  a  total  output  of  36,000  k.w.  at  120,000  volts.  There  would 
be  practically  no  new  or  experimental  engineering  work  in  con- 
nection with  such  a  scheme. 

Electrical  engineers  on  the  American  continent  are  rather  in- 
clined to  the  belief  that  when  energy  has  to  be  transmitted  from 
one  place  to  another  the  one  and  only  course  open  to  them  is  to 
adopt  the  three-phase  alternating-current  system.  It  is  not  sug- 
gested that  at  the  present  time  this  may  not,  in  the  majority  of 
cases,  be  the  best  system  available;  but  undoubtedly  there  are 
conditions  under  which  the  continuous-current  series  system 
would  prove  more  economical  and  reliable.  Of  course,  first  cost 
of  plant  and  operating  charges  have  to  be  taken  into  account 
when  comparing  different  systems,  and  the  most  satisfactory 
way  of  doing  this  is  to  reduce  all  estimated  costs  to  the  com- 
mon basis  of  annual  charges.  The  cost  of  the  direct-current 
generators  must  be  set  against  the  combined  cost  of  alternators 
and  exciters  and  step-up  transformers  with  all  intermediate 
switch  gear. 

In  this  connection  the  writer  cannot  refrain  from  quoting  a 
paragraph  which  occurred  in  one  of  the  leading  articles  in  the 
Electrical  World  of  New  York,  in  which  reference  is  made  to  the 
fact  that  transmission  by  continuous  currents  has  received  con- 
siderable attention  in  Europe. 

"Any  engineer  who  wanders  through  one  of  the  large  Thury 
stations  and  then  calls  to  mind  the  usual  long  concrete  catacombs 
bristling  with  high-tension  insulators  and  filled  with  dozens  of  oil 
switches,  scores  of  disconnecting  switches,  webbed  with  hundreds 
of  feet  of  high-tension  leads  and  spattered  with  automatic  cut- 
outs, will  stop  and  think  a  bit  before  he  complacently  sniffs  at 
high-tension  direct-current  transmission." 

In  regard  to  reliability  it  is  true  that,  on  the  Thury  system,  the 
generators  have  not  the  protection  against  lightning  disturbances 
which  the  step-up  transformers  afford  to  the  alternators  on  high- 
tension  three-phase  systems,  and  where  thunderstorms  are 
prevalent  this  must  not  be  overlooked,  as  the  cost  of  protective 
apparatus  may  prove  excessive.  In  this  connection  it  is  interest- 


244  ELECTRIC  POWER  TRANSMISSION 

ing  to  note  that  the  charges  of  electricity  in  the  upper  atmosphere 
are  always  positive,  and  the  negative  wire  will  therefore  tend  to 
draw  a  lightning  discharge  away  from  the  positive  wire  or  grounded 
guard  wire,  but  to  how  great  an  extent  this  would  affect  the 
proper  disposition  of  the  wires  it  is  difficult  to  say. 

This  chapter  will  be  concluded  with  a  brief  summary  of  the 
important  points  in  favor  of,  and  unfavorable  to,  the  employment 
of  continuous  currents  on  the  series  system  for  the  purpose  of 
transmitting  energy  at  comparatively  high  voltages  from  one 
place  to  another. 

ADVANTAGES  OF  THE  DIRECT-CURRENT  SERIES  SYSTEM 

1.  The  power-factor  is  unity — a  fact  which  alone  accounts  for 
considerable  reduction  of  transmission  losses. 

2.  Higher  pressures  can  be  used  than  with  alternating  cur- 
rent, the  conditions,  as  shown  by  actual  tests,  being  more  favor- 
able to  direct-current  transmission  than  is  generally  supposed. 
Without  any  alteration  to  insulation  or  spacing  of  wires,  ap- 
proximately double  the  working  pressure  can  be  used  if  direct 
current  is  substituted  for  alternating  current.     Moreover,  the 
insulation  is  subjected  to  the  maximum  pressure  only  at  times 
of  full  load,  whereas  on  the  parallel  system  the  insulation  is 
subject  to  the  full  electrical  stress  at  all  times. 

3.  There  is  no  loss  of  power  through  "dielectric  hysteresis" 
in  the  body  of  insulating  materials. 

4.  The  necessity  for  two  wires  only,  in  place  of  three,  effects  a 
saving  in  the  number  of  insulators  required  and  allows  cheaper 
line  construction. 

5.  Where  it  is  necessary  to  transmit  power  by  underground 
cables,  continuous  currents  have  great  advantages  over  alternat- 
ing currents.     Single-core  cables  can  be  made  to  work  with  con- 
tinuous currents  at  100,000  volts.     By  using  two  such  cables  and 
grounding  the  middle  point  of  the  system  it  is,  therefore,  quite 
feasible  to  transmit  underground  at  200,000  volts. 

6.  The  practicable  distance  of  transmission,  especially  when 
the  whole  or  a  part  is  underground,  is  greater  than  with  alternat- 
ing currents. 

7.  There  are  no  induction  or  capacity  troubles  and  no  surges 
or  abnormal  pressure  rises  due  to  resonance  and  similar  causes, 
such  as  have  been  experienced  with  alternating  currents.     This 


TRANSMISSION  BY  CONTINUOUS  CURRENTS  245 

virtually  makes  the  factor  of  safety  on  insulation  greater  than  on 
alternating-current  circuits,  even  when  the  working  pressure  is 
doubled. 

8.  A  number  of  generating  stations  can  easily  be  operated  in 
series,  and  when  the  demand  for  power  increases,  a  new  generating 
station  can  be  put  up  on  any  part  of  the  line  if  it  is  inconvenient 
to  enlarge  the  original  power  station. 

9.  The  simplicity  and  relatively  low  cost  of  the  switch  gear  is 
remarkable.     A    switch    pillar    with  ammeter,   voltmeter,  and 
four-point  switch  is  all  the  necessary  equipment  for  a  generator. 
The  switch  pillar  for  a  motor  includes,  in  addition,  an  automatic 
"by-pass"  which  bridges  the  motor  terminals  in  the  event  of  an 
excessive  pressure  rise.     This  compares  very  favorably  with  the 
ever-increasing — though  in  some  cases  unnecessary — complica- 
tion and  high  cost  of  the  switching  arrangements  in  high-tension 
power  stations  on  the  parallel  system. 

10.  With    the    Thury  system   any  class  of  supply  can  be 
given,    and  the  motors  can  be  made  to  drive  sub-generators 
capable  of  running  in  parallel  with  any  local  electric  generating 
plant. 

11.  In  hydraulic  generating  stations  where  the  variations  of 
head  are  considerable,  as  will  generally  be  the  case  if  there  is 
no  storage  reservoir,  a  greater  all-round  efficiency  can  be  obtained 
than  if  the  machines  had  to  be  driven  at  constant  speed. 

12.  For  any  industrial  operation  requiring  a  variable-speed 
drive  at  constant  torque,  the  Thury  motor,  without  constant- 
speed  regulator,  is  admirably  adapted.     It  might  have  a  useful 
application  in  the  driving  of  generators  supplying  constant  cur- 
rent to  electric  furnaces  in  which  the  voltage  across  electrodes 
is  continually  varying. 

DISADVANTAGES  OF  THE  DIRECT-CURRENT  SERIES  SYSTEM 

1.  The  necessity  of  providing  insulating  floors  and  mounting 
all  current-carrying  machines  and  apparatus  on  insulators.     The 
highly  insulated  coupling  required  to  transmit,  mechanically, 
large  amounts  of  power  between  prime  mover  and  electric  gener- 
ator is  also  objectionable. 

2.  The  smallness  of  the  generators;  the  output  of  each  gener- 
ator being  limited  by  the  line  current  and  the  permissible  voltage 
between  the  collecting  brushes  on  the  commutator.     One  prime 


246  ELECTRIC  POWER  TRANSMISSION 

mover  is  usually  coupled  to  two  or  more  direct-current  generator 
units.  This,  however,  is  necessarily  more  costly  than  if  larger 
electric  generators  could  be  used;  moreover,  it  practically  limits 
the  choice  of  hydraulic  turbines  to  the  horizontal  type,  since  the 
coupling  of  several  generators  on  the  shaft  of  a  vertical  water- 
wheel  would  be  difficult  and  unsatisfactory. 

3.  With  constant  current  on  the  line,  the  line  losses  are  the 
same  at  all  loads,  and  the  percentage  power  loss  in  conductors  is 
inversely  proportional  to  the  load.     This  is  exactly  the  reverse  of 
what  occurs  on  the  alternating-current  parallel  system,  in  which 
the  percentage  line  loss  is  directly  proportional  to  the  load.     It 
should,  however,  be  mentioned  that  on  the  Thury  system  the  line 
current  may  be  reduced  about  30  per  cent,  at  times  of  light  load, 
except  when  the  circuit  feeds  motors  of  industrial  undertakings 
requiring  constant  current  day  and  night.     It  must  not  be  over- 
looked that  large  percentage  losses  at  times  of  light  load  are  of 
serious  moment  only  where  steam  engines  are  used  or  where  stor- 
age reservoirs  are  provided  for  water-power  generating  stations. 
In  the  case  of  water-power  schemes  without  storage,  the  fact  of 
the  full-load  line  losses  continuing  during  times  of  light  load  is  not 
objectionable. 

4.  The  series  system  is  less  suitable  than  the  parallel  system 
for  distribution  of  power  in  the  neighborhood  of  the  generating 
station.     It  is  essentially  a  transmission  system,  not  a  distrib- 
uting system. 

5.  The  water  turbine  working  under  constant  head  is  not  the 
ideal  engine  for  driving  constant-current  machines. 

6.  Special    regulating     devices   are    necessary   to   maintain 
constant  speed  on  the  motors. 

7.  It    is    impossible    to  overload  the  motor,  even  for  short 
periods.     This  would  be  a  very  serious  objection  to  the  use  of 
these  motors  in  connection  with  electric  traction  systems. 

8.  Greater    liability    to    damage  and  interruption  from  the 
effects  of  lightning.     It  may  be  said  that  an  overhead  line, 
whether  for  alternating  or  direct  current,  is  always  liable  to  dam- 
age by  lightning;  but  with  the  high-tension  alternating-current 
system,  the  transformers  and  automatic  oil  switches  will  usually 
protect  the  generators  themselves  from  serious  damage,  while 
with  the  Thury  system  there  is  always  a  path  for  lightning  dis- 
charges through  the  generators  and  motors,  and  the  damage  done 
may  be  very  great.     This  simply  means  that  particular  attention 


TRANSMISSION  BY  CONTINUOUS  CURRENTS  247 

should  be  given  to  the  question  of  lightning  protection  on 
overhead  direct-current  lines,  and  the  ease  with  which  highly 
inductive  choke  coils  can  be  introduced  on  a  direct-current  system, 
without  opposing  any  obstacle  (except  ohmic  resistance)  to 
the  passage  of  the  line  current,  tends  toward  the  attainment  of 
increased  safety. 


CHAPTER  IX 

MECHANICAL  PRINCIPLES  AND  CALCULATIONS- 
OVERHEAD  CONDUCTORS 

123.  Introductory. — If  a  wire  is  stretched  between  two  fixed 
points,  A  and  B,  lying  in  the  same  horizontal  plane,  and  sepa- 
rated by  a  distance  of  I  ft.,  there  will  be  a  certain  sag  of  s  ft. 
in  the  wire.  This  sag  or  deflection  from  the  horizontal  line 
AB,  will  be  greatest  at  the  center  of  the  span,  and  its  value, 
for  a  given  length  of  span  (I  ft.),  will  depend  upon  the  weight  of 
the  wire  and  the  tension  with  which  it  has  been  drawn  up.  If 
the  wire  were  perfectly  uniform  in  cross-section  and  perfectly 
flexible,  the  curve  ADB  (Fig.  88)  would  be  a  catenary.  It  should 
be  observed,  however,  that  in  this  and  subsequent  diagrams,  the 
sag  s  is  shown  much  larger  relatively  to  the  span  /  than  it  would 


D 

FIG.  88. — Wire  hanging  between  two  supports  at  the  same  elevation. 

be  on  most  practical  transmission  lines;  and  as  the  span  I  is 
generally  very  little  shorter  than  the  length  of  the  wire  between 
the  suspension  points  A  and  B,  no  appreciable  error  is  introduced 
by  assuming  the  weight  of  the  wire  to  be  distributed  uniformly 
along  the  horizontal  line  ACB  instead  of  along  the  curve  ADB. 
On  this  assumption  the  curve  A DB  becomes  a  parabola,  and  as  the 
calculations  are  more  easily  made  on  the  assumption  of  a  para- 
bolic curve  than  with  the  possibly  more  correct  catenary,  it  is 
customary  to  use  the  formulas  relating  to  the  parabola  for  the 
solution  of  sag-tension  problems.  On  very  long  spans,  even 
with  fairly  large  conductors,  the  actual  curves  assumed  by  the 
wires  hanging  in  still  air  under  the  influence  of  their  own  weight 
only,  will  approximate  more  nearly  to  the  catenary  than  to  the 

248 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    249 


parabola,  and  if  it  is  desired  to  introduce  refinement  of  calcu- 
lation, the  catenary  is  the  more  correct  curve  to  work  to  on  long 
spans,1  but  owing  to  the  many  more  or  less  arbitrary  assumptions 
that  must  necessarily  be  made  in  all  calculations  on  the  mechan- 
ical features  of  a  transmission  line,  such  refinements  are,  in  the 
writer's  opinion,  unnecessary  and  only  justifiable  when  they  in- 
volve no  additional  time  or  labor  in  the  calculations. 

124.  Graphical  Statics  Applied  to  Transmission-line  Calcula- 
tions— General  Problem. — Instead  of  assuming  a  definite  shape 
of  curve  to  represent  the  form  taken  by  a  wire  which  is  free  to 
hang  between  two  supports  a  known 
distance  apart,  it  is  proposed  to  de- 
velop the  necessary  formulas  for  sag 
and  tension  calculations  by  applying 
the  well-known  principles  of  graphical 
statics. 

Consider  a  mass  of  any  irregular 
shape,  the  weight  of  which  may  be 
represented  by  the  vertical  vector  OPa 
passing  through  its  center  of  gravity 
O'  as  shown  in  Fig.  89.  This  mass 
is  suspended  by  two  perfectly  flexible 
ties  from  the  fixed  points  A  and  B. 
Except  for  the  special  case  of  parallel 
forces,  the  resultant  PG  and  the  com- 
ponents PA  and  PB  acting  in  the 
direction  of  the  suspension  cords,  AC 
and  BD,  will  pass  through  the  com- 
mon point  0.  The  system  of  forces 

is  in  equilibrium  and  the  conditions  to  be  fulfilled  are  there- 
fore, 

1.  That  the  vectorial  sum  of  all  forces  and  reactions  shall  be 
zero,  and 

2.  That  the  sum  of  all  moments  taken  about  any  point  shall 
be  zero. 

1  In  his  paper  read  at  the  annual  Convention  of  the  A.  1.  E.  E.  at  Chicago, 
June,  1911,  Mr.  W.  Le  Roy  Robertson  works  out  solutions  of  sag  and  span 
problems  with  the  aid  of  the  catenary.  A  more  recent  contribution  by 
Mr.  F.  K.  Kirsten  in  Trans.  A.  I.  E.  E.,  p.  735,  vol.  xxxvi,  1917,  contains  a 
complete  mathematical  analysis  of  the  mechanical  problems  on  the  assump- 
tion that  the  wires  take  the  form  of  the  catenary  curve. 


89. — Mass     suspended 
ro    points — diagram   of 


250  ELECTRIC  POWER  TRANSMISSION 

The  following  is  an  explanation  of  the  manner  in  which  a 
force  diagram  similar  to  Fig.  89  may  be  constructed. 

Known  Data. — 1.  The  position  of  suspension  points  A  and  B. 
1  2.  The  magnitude  and  position  relatively  to  points  A  and  B 
of  the  force  of  gravity  (i.e.,  the  length  of  the  vector  POG  and  the 
distance  AE  of  this  vector  from  the  point  A). 

3.  The  magnitude,  but  not  the  direction,  of  the  force  PB 
acting  through  the  suspension  point  B. 

Required. — 1.  The  direction  of  the  force  PB- 

2.  The  magnitude  and  direction  of  the  force  PA. 

3.  The  horizontal  and  vertical  components  of  the  forces  at 
the  points  of  support. 

Taking  moments  about  the  point  A  gives  the  equation 

PB  X  AR  =  PG  X  AE 
thus' 

AR  =  AE  X  ^ 

"B 

From  the  point  A  as  a  center,  use  this  radius  to  draw  an  arc 
of  circle  the  tangent  to  which,  passing  through  the  point  B,  will 
locate  the  point  O.  Draw  OPG  to  the  proper  scale  to  represent 
the  force  of  gravity  and  complete  the  parallelogram  of  forces. 
The  vertical  component  of  the  reaction  at  point  A  is  NO,  and  at 
the  point  B  it  is  MO.  The  horizontal  reactions  at  the  points  of 
support  are  PAN  and  PBM  respectively.  These  are  obviously 
equal,  but  opposite  in  direction,  in  every  conceivable  case  of  a 
body  in  equilibrium  subject  only  to  the  force  of  gravity  acting, 
as  it  always  does,  vertically  downward. 

125.  Stretched  Wire.  Supports  on  Same  Level. — In  Fig.  90, 
a  wire  weighing  w  Ib.  per  foot  is  stretched  between  the  supports 
A  and  B  lying  in  the  same  horizontal  plane.  The  maximum 
tension  in  the  wire  is  PB  Ib.  This  is  the  tension  at  the  points  of 
support,  and,  owing  to  the  symmetry  of  the  figure,  it  is  the  same 
in  amount  at  A  as  at  B.  The  assumption  is  now  made  that  the 
total  weight  of  wire  is  equal  to  the  weight  per  foot  multiplied  by 
the  straight  line  distance  between  the  points  A  and  B.  Thus 

PG  =  w  XI 

where  I  is  the  distance  in  feet  between  A  and  B.  This  assump- 
tion is  allowable  on  all  except  spans  of  extraordinary  length, 
because  the  actual  length  of  the  wire  differs  only  by  a  very  small 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    251 

amount    from    the   shortest   distance    between    the   points    of 
suspension. 

Draw  the  vector  OPG  to  represent  the  total  downward  force 
wl.  It  will  lie  on  a  vertical  line  midway  between  A  and  B. , 
Let  it  be  bisected  by  any  horizontal  line  such  as  AB.  From  0 
lay  off  OPB  at  such  an  angle  that  the  head  PB  of  the  vector  lies 


FIG.  90. — Diagram  of    forces — suspended  wire    with    supports  on  same  level. 

on  the  horizontal  line  bisecting  OPG.  Complete  the  parallelo- 
gram of  forces.  Then  ON  represents  the  vertical  component  at 
each  point  of  support,  and  NPB  or  NPA  is  the  horizontal  force 
PH  acting  at  each  support;  it  is  also  the  total  tension  in  the  wire 
at  the  center  or  lowest  point  of  the  span. 

Referring  to  Fig.  91,  which  shows  a  span  of  length  I,  we  can 
now  calculate  the  sag  s  at  the  center.     Consider  the  moments 


Fio.  91.  —  Diagram    of    forces    for    sag    calculation  —  supports    on    same    level. 

about  the  point  A  due  to  the  half  span  AD.     These  must  balance, 
and  the  equation  is 


whence 


8Ph 


(101) 


which  is  the  well-known  formula  for  sag  calculations  when  the 
parabolic  assumption  is  made. 

Given  an  overhead  conductor  hanging  in  still  air  from  supports 


252 


ELECTRIC  POWER  TRANSMISSION 


at  the  same  level,  under  the  influence  of  its  own  weight  only, 
the  tension  in  the  wire  at  the  center  of  the  span  will  therefore 
be 

P*=f  (102) 

and  in  practical  work,  except  perhaps  on  exceptionally  long 
spans,  the  maximum  tension  in  the  wire  will  be  so  nearly  equal 
to  the  horizontal  component  of  the  pull  that  the  tension  at  the 
points  of  support  (where  the  maximum  pull  occurs)  may  be  as- 
sumed the  same  as  the  horizontal  component  as  calculated  by 
formula  (102).  The  tension  at  any  point  in  the  span  can,  how- 
ever, be  very  easily  calculated.  Thus,  let  0  in  Fig.  92  be  the 
angle  which  the  tangent  to  the  curve  makes  with  the  horizontal 


FIG.  92. — Vector  diagram  of  forces  for  calculating  tension  at  any  point  in  the 
span. 

at  any  point  0  distant  y  feet  from  the  vertical  plane  normal  to 
the  wire  at  the  lowest  point.  On  the  assumption  previously 
made  that  the  weight  is  distributed  uniformly  over  the  straight- 
line  distance  between  the  two  points  of  support,  the  weight  of 
the  section  considered  in  Fig.  92  is  y  X  w  where  w  is  the  weight 
of  the  conductor  per  foot  length.  This  acts  vertically  downward 
at  the  point  0,  and  we  may  write 

tan  6  =  -~ 

Substituting  for  Ph  the  value  given  by  formula  (102),  we  have, 
tan  6  =  y  X  ~  (103) 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS     253 

which  becomes  a  maximum  at  the  points  of  support,  where  its 
value  is 

tan  0  =  -y  (104) 

The  wire  will  take  the  stress  in  the  direction  of  the  tangent  to 
the  curve,  and  the  resultant  tension  at  the  point  0  is  therefore 

OP0 


=  wy  X  cosec  0  (105) 

which  is  readily  solved  with  the  aid  of  trigonometrical  tables, 
since  the  angle  0  is  given  by  formula  (103). 

The  use  of  tables  can  be  avoided  by  putting  the  expression  in 
the  form 


Po  =  ys       l*  +  64  «V  (106) 

the  maximum  value  of  which  occurs  at  the  points  of  suspension  of 
the  wire  where  it  becomes 

»..7 

(107) 

A  simpler  formula,  which  is  a  very  close  approximation  to  the 
correct  formula,  is 

Pmax  =  Ph  +  uwf. 


ws 


1  This  formula  is  based  on  the  fact  that  when,  the  quantity  a  is  small 
relatively  to  A,  it  is  permissible  to  write, 


2A 

Thus,  the  total  or  resultant  force  at  the  point  of  support  is, 


wly, 
wl*  . 


254  ELECTRIC  POWER  TRANSMISSION 

The  ratio  of  the  maximum  tension  (at  point  of  support)  to 
the  tension  at  center  of  span  is, 

Pnax  =  Formula(107)  = 
Ph        Formula(l02) 

8s2 
which,  approximately  =  1  +  —  (110) 

Seeing  that  5  is  usually  very  small  in  relation  to  /,  the  quantity 
given  by  formula  (109)  or  (110)  is  generally  so  nearly  equal  to 
unity  that  no  serious  error  is  introduced  by  using  the  simple 
formula  (102)  for  calculating  the  tension  in  the  conductors; 
in  other  words,  the  wire,  as  previously  mentioned,  is  so  nearly 
horizontal  throughout  its  entire  length  that  the  horizontal 
component  of  the  tension,  instead  of  the  resultant,  may  be 
considered  as  the  tension  acting  at  any  point  on  the  wire. 

The  length  of  the  parabolic  curve  ADB  (Fig.  88)  is, 


but  it  is  usual  to  omit  all  except  the  first  two  terms  of  the  series. 
This  gives, 


126.  Supports  at  Different  Elevations.  —  Many  students  and 
some  engineers  appear  to  have  trouble  in  understanding  the 
distribution  of  forces  in  overhead  lines  carried  up  a  steep  grade: 
the  idea  that  the  poles  carry  considerably  increased  loads  as 
they  occupy  positions  higher  up  the  hill-side  is  not  uncommon, 
and  is  sometimes  put  forward  in  explanation  of  the  troubles 
which  are  not  infrequent  with  poorly  constructed  lines  carried  up 
steep  inclines.  Such  troubles  as  have  occurred  in  the  past  were 
perhaps  not  entirely  unconnected  with  the  fact  that  the  design- 
ing engineer,  or  the  construction  engineer,  or  both,  had  no  clear 
understanding  of  the  distribution  of  forces  in  such  a  line.  The 
problem  is  very  simple  if  the  component  forces  necessary  to  the 
state  of  equilibrium  are  studied  as  in  the  preceding  article  wherein 
the  supports  were  supposed  to  be  at  the  same  elevation. 

In  Fig.  93,  the  difference  in  elevation  of  the  supports  is  h 
feet.  The  span  measured  horizontally  is  I  feet,  and  the  straight- 
line  distance  between  the  supports  A  and  B  is  I'  feet.  If  6 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    255 

is  the  angle  which  the  line  AB  makes  with  the  horizontal,  we 
may  write 

tan  6  = 


and 


cos  e  =  j 


The  weight  of  wire  per  foot  is  w  lb.,  and  it  is  assumed  that  the 
total  weight  is  wl'  lb.  This  force  acts  through  the  point  C  mid- 
way between  A  and  B.  The  other  known  quantity  is  the  mag- 
nitude (but  not  the  direction)  of  the  maximum  tension  in  the 
wire:  this  is  the  force  PB  acting  through  the  highest  point  of 
support  (B}. 


FIG.  93. — Diagram  of  forces — supports  at  different  elevations. 

Draw  the  vertical  line  OP0  to  represent  the  total  force  of 
gravity  (wl'}.  Through  its  center  C  draw  the  line  AB  (or  a  line 
parallel  to  AB}  making  an  angle  6  with  the  horizontal.  Then 
from  O  as  a  center  describe  an  arc  of  radius  OPB  equal  to  the 
known  force  PB.  Its  intersection  with  AC  locates  the  head  of 
the  vector  OPB.  Complete  the  parallelogram  of  forces,  and 
drop  the  perpendiculars  PAM  and  PBN  on  to  the  vertical  OPG. 
These  are  a  measure  of  the  (equal)  horizontal  components 
of  the  reactions  at  the  two  points  of  support,  and  also  of  the  total 
tension  in  the  wire  at  the  lowest  point  D.  The  length  MO  is  the 
vertical  component  of  the  reaction  at  the  lower  support  A ;  while 
NO  is  the  vertical  reaction  at  B.  Their  sum  is,  of  course,  equal 
to  OPG.  The  angle  ft  which  the  vector  PB  makes  with  the 
horizontal  is  the  slope  of  the  wire  where  it  leaves  the  point  B, 


256  ELECTRIC  POWER  TRANSMISSION 

The  construction  as  above  described  satisfies  the  fundamental 
conditions  of  equilibrium,  namely,  that  all  vertical  forces  shall 
balance;  that  all  horizontal  forces  shall  balance,  and  that  the  sum 
of  all  moments  taken  about  any  point  shall  be  equal  to  zero. 

In  practice  it  is  not  convenient  to  solve  such  problems  by 
actual  measurement  of  quantities  plotted  to  scale  on  drawing 
paper,  because  the  horizontal  components  of  the  forces  are  usually 
very  much  greater  than  the  vertical  components.  A  trigo- 
nometrical solution  is  therefore  desirable. 

In  Fig.  93,  the  known  quantities  are  the  two  sides  OPB  and  OC 
of  the  triangle  OCPB  of  which  the  angle  OCPB  is  also  known, 
being  equal  to  0  +  90  degrees. 

The  angle  OPBC  or  <p  may  be  calculated  from  the  relation 

sin  <p  =  -p-  sin  (90  +  0)  =  ^-  cos  0 

wl 
~2PB 

and  the  angle  ft  which  the  wire  at  the  upper  support  makes  with 
the  horizontal  is  therefore 

/   ...7  » 

(112) 

With  the  aid  of  trigonometrical  tables  or  a  slide  rule,  the  values 
of  sin  ft  and  cos  ft  can  be  obtained,  and  the  other  components  of 
the  force  diagram  readily  calculated.  Thus,  the  horizontal  com- 
ponent at  either  point  of  support,  which  is  also  the  total  tension 
in  the  wire  at  the  point  (if  any)  where  the  slope  is  zero,  is 

Ph  =  PB  cos  ft  (113) 

The  vertical  component  of  the  reaction  at  the  highest  point  B, 
where  this  reaction  is  greatest,  is 

PBV  =  NO  =  PB  Sin  ft  (114) 

The  weight  supported  by  the  lower  pole  is 

PAV  =  MO  =  wl'  -  PBV  (115) 

This  may,  obviously,  be  a  negative  quantity,  i.e.,  the  wire  may 
exert  an  upward  pull  on  the  lower  support.  In  that  case  the  sag  s 
below  the  point  A,  will  be  zero,  and  this  would  correspond  to  the 
usual  condition  on  a  steep  incline,  or  on  a  moderate  incline  if  the 
spans  are  short. 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    257 

Position  of  Lowest  Point  of  Span.  Supports  at  Different  Eleva- 
tions. —  The  position  of  the  lowest  point  D  in  the  span  will  be  deter- 
mined by  the  vertical  weight  carried  by  each  of  the  two  supports. 
Thus,  referring  to  Fig.  93,  the  vertical  component  of  the  forces 
acting  at  B  is  simply  the  weight  of  that  portion  of  the  wire 
comprised  between  the  support  B  and  the  point  D  where  the 
tension  in  the  wire  has  no  vertical  component.  The  horizontal 
distance  of  the  point  D  from  B  (in  feet)  is 


=    "  (H6) 

This  may  give  a  distance  for  1B  which  is  greater  than  I.  In  that 
case  the  support  A  would  be  the  lowest  point  in  the  span.  The 
horizontal  distance  from  the  lower  point  of  support,  A,  to  the 
point  D  is  1A  =  I  —  IB',  which  shows  that  1A  will  be  a  negative 
quantity  when  1B  is  greater  than  I. 

If  it  is  desired  to  express  these  formulas  in  terms  of  the  hori- 
zontal component  (Ph)  of  the  tension,  a  reference  to  the  force 
diagram  in  Fig.  93  will  show  that  we  may  write 

PG  =  2  (PBV  -  Ph  tan  6) 


the  upper  support  is 

wl' 
PBv  =         +  Ph  tan  6 


Substituting  in  formula  (116),  we  have: 
Similarly 


0.9) 


These   formulas    can    be   solved   without  reference  to  trigono- 
metrical tables,  because,  although  I'  =  --  -j  it  can  also  be  ex- 

COS    v 

pressed  as  V  =  \/l2  +  h2. 

17 


258  ELECTRIC  POWER  TRANSMISSION 

Formula  (119)  shows  that  if  — p  is  equal  to  ^»    the    lowest 

point  of  the  wire  coincides  with  the  lower  support  A,  while,  if 
the  second  term  in  the  equation  is  greater  than  the  first,  1A 
is  negative,  and  there  is  a  possibility  of  the  resultant  upward 
pull  on  the  lower  insulator  at  A  being  greater  than  the  downward 
pull  due  to  the  weight  of  the  wires  in  the  adjoining  span.  It  is 
well  to  bear  this  point  in  mind  when  considering  an  abrupt 
change  in  the  grade  of  a  transmission  line. 

127.  Calculation  of  Sag  with  Supports  on  an  Incline. — The 
formula  (101)  as  calculated  for  spans  with  supports  on  the  same 
level  may  be  used  for  spans  on  an  incline  provided  the  distance 
1B  of  Fig.  93  is  considered  as  half  of  a  level  span  of  which  the  sag 
is  (s  +  h).  Thus, 

CH-*)-2pf~h  (120) 

Similarly 

...72 

(121) 


2Ph  cos  9 

On  a  steep  incline,  where  A  is  the  lowest  point  of  the  span, 
it  is  more  useful  to  know  the  maximum  deflection  of  the  wire 
from  the  straight  line  AB  as  observed  by  sighting  between  the 
points  A  and  B.  This  maximum  deflection  will  occur  at  the 
center  of  the  span.  A  careful  study  of  the  force  diagram  in  Fig. 
93  will  make  it  clear  that,,  just  asOPB  is  the  slope  of  the  tangent  to 
the  wire  at  the  point  B,  and  PBN  is  the  slope  of  the  tangent  to 
the  wire  at  the  lowest  point  D,  so  PBC  is  the  slope  of  the  tangent 
to  the  curve  at  the  middle  of  the  span.  This  is  therefore  the 
point  of  maximum  deflection  from  the  straight  line  AB. 

Consider  now  Fig.  94.  The  maximum  deflection  of  the  line 
on  the  slope  is  sf,  and,  by  taking  the  sum  of  all  the  moments  about 
A  and  putting  this  sum  equal  to  zero,  the  value  of  this  deflection 
is  found  to  be, 


(I22) 

which  indicates  that  the  maximum  deflection  from  the  straight 
line,  of  a  conductor  strung  between  supports  on  a  slope,  as  meas- 
ured at  the  center  of  the  span,  is  exactly  the  same  as  the  maxi- 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    259 


mum  sag  s  of  the  same  conductor  strung  between  points  on  the 
same  level;  provided  the  span  measured  horizontally  and  the 
horizontal  component  (PA)  of  the  tension  are  the  same  in  both 
cases. 


If  P  is  the  tension 
(122)  may  be  written 


\COS  0  ' 


in  the  direction  AB,  the  formula 


wl* 


Length  of  Wire  with  Supports  at  Different  Elevations.- — The  length 
of  wire  between  the  two  supports  A  and  B  may  be  considered  as 
the  sum  of  two  distinct  portions  of  the  parabolic  curve  ADB 


wl' 


FIG.  94.  —  Diagram  of  forces  for  sag  calculation  —  supports  at  different  elevations. 


(Fig.  93)  ;  the  one,  AD,  the  length  of  which,  according  to  formula 

(111),  is 


'   3  X  2k 

and  the  other,  DB,  of  length 

\B  =  1B  +    3  x  2JB 

The  sum  of  these  two  quantities  is 

x  =  ;  +  2r(1+«!  + 

o  L.        la 


(123) 


128.  Example  Illustrating  Use  of  Formulas. — Consider  a  span 
of  485  ft.  measured  horizontally,  with  a  difference  of  level  of  40  ft. 
between  points  of  support.  The  wire  is  No.  2-0  B.  &  S.  gauge 
aluminum,  weighing  647  Ib.  per  mile.  Calculate  vertical  and 
horizontal  components  of  forces,  also  sag  and  position  of  lowest 
point  of  span,  if  the  maximum  tension  in  the  wire  is  185  Ib. 


260  ELECTRIC  POWER  TRANSMISSION 

The  known  quantities  are  : 

I  =  485 
h  =  40 
w  =  0.1226 
PB  =  185 
The  unknown  quantities  are  calculated  as  follows: 

tan  e  =  ^=  =  0.0825 
4o5 

9  =  4°  43' 
cos  0  =  0.9966 

485X0.1226       _  1ft1 
Sm^=       2X185       =0'161 

V  =  9°  15' 

By  formula  (112)  ft  =  6  +  <f>  =  13°  58' 
cos  ft  =  0.971 
sin  ft  =  0.2413 

By  (113)  Ph  =  185  X  0.971  =  180  Ib. 

By  (114)  PBV  =  185  X  0.2413  =  44.6  Ib. 


By  (116)  t--  364  ft. 


Thus  the  sag  below  the  bottom  support  will  be  5.2  ft.  when  the 
maximum  tension  is  185  Ib. 

If  it  is  desired  to  avoid  the  use  of  trigonometrical  tables,  the 
procedure  would  be  as  follows;  but  the  assumption  now  made  is 
that  the  known  tension  is  the  horizontal  component,  or  Ph  =  180  Ib. 

The  straight-line  distance  between  the  two  points  of  support 
is  V  =  VJ^TT2  =  V(485)2  +  (40)  2  =  486.5  ft.  (approx.). 

By  formula  (117) 

0.1226  X  486.5    .   40  X  180 
PBV  =          ~2~  —  485— 

By  formula  (118) 

40  V  180 

=  363  f 


0.1226X486 

and 

1A  =  485  -  363  =  122  ft. 
The  sag  s  measured  from  the  lowest  support  is  given   by 

formula  (121)  wherein  cos  0  can  be  replaced  by  p-     Thus, 
_  0.1226  (122)2  X  486.5 
2  X  180  X  485 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    261 

which  is  probably  more  accurate  than  the  dimension  5.2  obtained 
in  the  preceding  calculation  by  subtracting  h  from  the  total 
sag  (measured  from  the  upper  support).  No  high  degree  of 
accuracy  is  claimed  for  the  results  of  these  numerical  examples, 
the  calculations  being  made  on  the  slide-rule. 

129.  Conclusions — Overhead  Lines  on  Steep  Grade. — Except 
for  the  fact  that  the  pole  foundations  must  be  wisely  chosen,  and 
sp.ecial  attention  paid  to  the  setting  of  the  poles,  there  are  no 
engineering  difficulties  in  the  way  of  running  an  overhead  line 
up  a  steep  incline.     It  may  be  well  to  reduce  the  length  of  span 
measured  horizontally  so  that  this  is  equal  to  about  V  cos  6, 
where  V  is  the  distance  between  supports  on  level  ground,  and  6  is 
the  angle  which  the  average  slope  of  the  line  makes  with  the 
horizontal.     This  will  ensure  that  the  vertical  load  to  be  sup- 
ported by  the  insulators  does  not  exceed  the  vertical  load  on  the 
level  runs;  but  no  difficulty  need  be  experienced  in  arranging 
for  each  pole  to  take  its  proper  share  of  the  weight  of  the  wire 
(with  or  without  ice  loading). 

The  stringing  of  the  wires  may  be  done  with  the  aid  of  a  dyna- 
mometer; or  the  deflection,  as  calculated  by  formula  (122),  maybe 
measured  by  sighting  from  pole  to  pole.  If  the  poles  or  suspen- 
sion insulators  of  a  line  carried  on  an  incline  are  deflected  from  the 
vertical  owing  to  the  unbalancing  of  horizontal  forces,  at  the 
time  when  the  wires  are  strung,  the  effect  must  be  attributed  to 
incompetence  on  the  part  of  the  engineer  in  charge  of  construction, 
and  not  to  any  unexplained  flaw  in  the  known  laws  of  equilib- 
rium. If  the  conductor  is  tied  to  the  insulator  at  the  proper 
point,  the  horizontal  components  of  the  forces  at  this  point  will 
balance,  and  the  only  forces  which  the  insulators  will  have  to 
resist,  when  the  conductors  are  hanging  under  normal  conditions 
in  still  air,  are  those  due  to  gravity  acting  vertically  downward : 
there  should  be  no  unbalanced  forces  tending  to  move  the  point 
of  support  in  the  direction  of  the  line,  either  up  or  down  the 
hill-side. 

130.  Effect  of  Temperature  Variations  on  Overhead  Wires.— 
If  a  is  the  temperature  coefficient  of  the  material  of  the  conductor 
(see  the  table  of  constants  in  Chapter  IV,  page  75),  then  the 
length  of  the  wire  when  the  temperature  is  raised  t  degrees  is, 

X,  =  Xo  (1  +  at)  (124) 

in  which  X0  is  the  original  length  of  the  wire. 


262  ELECTRIC  POWER  TRANSMISSION 

This  formula  assumes  that  the  wire  is  unstressed,  or  that  the 
stress  remains  unaltered  notwithstanding  the  increase  in  tem- 
perature. This,  however,  is  not  the  case  with  an  overhead  con- 
ductor. As  indicated  by  formula  (102),  the  tension  in  a  wire  sus- 
pended horizontally  between  two  fixed  supports  is  almost  exactly 
proportional  to  the  square  of  the  span,  and  inversely  proportional 
to  the  sag  at  center  of  span.  The  effect  of  temperature  variation 
is  to  alter  the  length  of  wire  and  therefore  the  amount  of  sag 
and  tension.  With  a  reduction  of  temperature,  the  length  of 
wire  will  decrease;  this  will  cause  an  increase  in  the  tension, 
but  owing  to  the  fact  that  the  wire  will  stretch  under  the  influence 
of  the  increased  tension,  the  sag  at  the  lower  temperatures  will 
be  somewhat  greater  than  it  would  be  if  there  were  no  elonga- 
tion of  the  wires  with  increase  of  stress.  All  sag-temperature 
calculations,  whatever  the  method  adopted,  must  therefore  take 
into  account  not  only  the  effect  of  elongation  with  increase  of 
temperature,  but  also  the  effect  of  the  elastic  contraction  of  the 
wire  with  increase  of  sag. 

If  P  is  the  tension  in  a  wire  of  cross-section  A,  and  if  M  is  the 
elastic  modulus  (as  given  for  various  materials  in  the  Table  of 
Article  41,  Chapter  IV,  p.  75),  then  the  elongation  of  the  wire 
due  to  the  tension  P  is, 


. 
Ae 


A  X  M 

the  original  length  of  the  wire  being  X. 

p 
If  instead  of  -r  the  letter  S  be  used  to  denote  the  stress  in 

pounds  per  square  inch  of  cross-section,  the  formula  becomes 
Xe  =  X  X  ^  (125) 

It  is  customary  to  assume  that  the  material  of  the  conductors 
is  perfectly  elastic  up  to  a  certain  critical  stress  known  as  the 
elastic  limit;  that  is  to  say,  if  the  application  of  a  certain  stress 
produces  a  strain  represented  by  Xe,  it  is  assumed  that,  on  the 
removal  of  the  stress,  the  conductor  will  contract  to  its  original 
length  X,  and  that  this  process  of  elongation  and  contraction 
follows  a  straight-line  law.  This  is  not  scientifically  correct, 
because,  on  removal  of  load,  the  amount  of  contraction  is  not 
directly  proportional  to  the  decrease  of  stress;  but  the  depart- 
ure from  the  straight-line  law  is  not  considerable,  and  no  serious 
error  is  introduced  by  disregarding  refinements  of  this  nature. 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    263 

A  matter  of  greater  importance  is  the  fact  that,  when  stranded 
conductors  are  used,  the  ratio  between  stress  and  strain  is  not 
correctly  given  by  the  modulus  M  as  calculated  from  tests  made 
on  solid  wires.  The  modulus  for  stranded  cables  will  depend 
upon  the  number  of  strands,  the  "lay"  of  the  strands,  whether 
the  central  core  is  of  metal  or  hemp,  etc.,  and  it  should  be  deter- 
mined by  actual  tests  on  samples  of  the  completed  cable;  but 
since  it  will  depend  largely  on  the  stress  to  which  the  cable  has 
been  subjected,  and  will  ultimately  differ  little  from  the  coeffi- 
cient for  solid  wires,  it  is  usual  in  sag  calculations  to  use  the  same 
value  of  M  for  both  solid  and  stranded  conductors.  Great 
accuracy  in  sag-temperature  calculations  is  only  necessary  in 
the  case  of  very  long  spans  (1000  feet,  and  over). 

A  very  simple  and  convenient  diagram  may  be  constructed, 
by  which  the  elongation  of  a  conductor  owing  to  change  of  stress 
or  temperature,  acting  either  singly  or  "jointly,  can  readily  be 
ascertained.  Fig.  95  shows  such  a  diagram,  constructed  for 
stranded  conductors  of  either  copper  or  aluminum.  Referring  to 
the  vertical  scales  from  left  of  diagram  to  center,  if  a  straight 
line  be  drawn  from  a  point  on  scale  1  corresponding  to  the 
stress  in  a  copper  conductor,  to  a  point  on  the  center  scale  cor- 
responding to  the  temperature,  the  point  at  which  this  line 
crosses  scale  2  will  indicate  the  increase  in  length  above  the  con- 
dition of  zero  stress  and  zero  temperature.  This  increase  in 
length  is  expressed  as  a  percentage,  such  as  feet  per  hundred  feet 
of  the  conductor. 

As  an  example,  an  increase  in  length  of  0.1  per  cent,  of  a  copper 
conductor  will  occur 

(a)  When  the  stress  remains  constant  and  the  temperature 
increases  104°. 

(b)  When  the  temperature  remains  constant  and  the  stress 
increases  15,000  Ib.  per  square  inch. 

(c)  When  the  stress  increases  from  12,000  to  15,500  Ib.  per 
square  inch  at  the  same  time  as  the  temperature  falls  from  137 
to  6°  F. 

The  vertical  scales  4  and  5  are  used  in  a  similar  manner  with 
the  common  temperature  scale  for  determining  the  changes  in 
length  of  stranded  aluminum  conductors. 

The  value  of  M  for  stranded  aluminum  conductors  which  has 
been  used  in  constructing  the  diagram  Fig.  95,  is  7,500,000  instead 
of  the  value  M  =  9,000,000  which  is  more  frequently  used  in  sag- 


264  ELECTRIC  POWER  TRANSMISSION 

temperature  calculations.  If  it  is  desired  to  use  the  larger  figure 
for  this  modulus,  the  stress  as  read  off  the  center  scale  for  Alumi- 
num, must  be  multiplied  by  1.2. 

(rantnrantv)   qaut  ajenbs  iad  -sqi-onx  issaxjg 

S  3  3  3  3  O  00  t-lOUJ^CO^^O 


aiq«0  ranntmniv  - 


iax  u; 


t  aaenbs  a»d 
FIG.  95. — Chart  for  calculating  changes  in  length  of  overhead  conductors. 

131.  Abnormal  Stresses  in  Wires  Due  to  Wind  and  Ice. — 
When  a  wire  hangs  between  horizontal  supports  in  still  air  under 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  265 

the  influence  of  its  own  weight  only,  the  tension  at  center  of  span 
for  a  definite  distance  between  supports  and  a  definite  sag  at 
center  will  be  proportional  to  the  weight  per  foot  length  of  the 
wire.  When  loaded  with  sleet  or  ice,  or  subject  to  wind  pressures, 
or  a  combination  of  both  these  additional  loads,  the  calculations 
of  the  proper  sag  to  allow  are  based  on  the  assumption  that  the 
weight  per  foot  is  no  longer  that  of  the  wire  only,  but  n  times 
this  amount;  where  n  is  a  factor  depending  upon  the  character 
and  amount  of  the  abnormal  loading  that  the  wire  is  likely  to  be 
subjected  to. 

Ice  and  Sleet. — The  effect  of  snow  and  sleet  adhering  to  the 
wires,  and  forming  an  ice  coating  of  variable  thickness,  is  to  add 
to  the  dead  weight  of  the  wires  and  offer  an  increased  surface  to 
wind  blowing  across  the  line.  Sleet  will  generally  collect  with 
slightly  greater  thickness  near  the  lowest  point  of  the  span, 
but  it  is  usual  to  assume  that  the  extra  vertical  loading  is  uni- 
formly distributed  over  the  whole  length  of  the  span.  Mr.  Max 
N.  Collbohm1  says  that  in  the  winter  of  1908-9,  in  Wisconsin  and 
neighboring  States,  the  snowstorm  of  January  28  covered  nearly 
all  overhead  wires  with  sleet  and  snow  2^  in.  to  4  in.  diameter. 
The  temperature  went  down  to  4°  F.  below  zero,  while  a  wind 
velocity  of  40  miles  per  hour  was  recorded.  These  conditions 
were,  of  course,  exceptional;  at  the  same  time  an  average  coat- 
ing of  sleet  and  snow  weighing  %  Ib.  per  foot  of  wire  is  not 
unusual.  A  coating  of  ice  ^  in.  thick  on  wires  running  through 
districts  where  sleet  and  low  temperatures  are  common  is  gener- 
ally allowed  for  in  calculations.  Sometimes  this  is  increased  to 
a  radial  thickness  of  %  in.  Sleet  deposits  6  in.  in  diameter 
have  actually  been  observed  on  some  of  the  steel  conductors  of 
the  Central  Colorado  Power  Company.  The  weather  condi- 
tions in  some  of  the  passes  through  which  this  line  is  carried  are, 
however,  particularly  severe. 

Taking  the  weight  of  sleet  at  57.3  Ib.  per  cubic  foot,  the  total 
weight  per  foot  of  the  loaded  conductor  is  w  +  1.25$  (d  +  £) 
where  w  is  the  weight  of  the  wire;  t,  the  radial  thickness  of  sleet 
(assumed  to  be  of  circular  section),  and  d,  the  diameter  of  the 
uncoated  wire;  these  dimensions  being  expressed  in  inches. 
Thus,  in  the  case  of  a  No.  OB.  &  S.  gauge  wire,  d  equals  0.325  in., 
and  assuming  t  equals  0.5  in.,  the  weight  of  the  ice  coating  is 
1.25  X  0.5  X  0.825  =  0.516  Ib.,  or  about  %  Ib.  per  foot  run, 
1  Electrical  World,  New  York,  March  25,  1909,  p.  734, 


266  ELECTRIC  POWER  TRANSMISSION 

which  is  a  common  allowance  to  make  for  ice  loads  on  wires  of 
average  size. 

Effect  of  Material  of  Conductor  on  Sleet  Deposits. — It  is  fre- 
quently contended  that  sleet  does  not  deposit  readily  on  alumi- 
num owing  to  the  greasy  character  of  the  oxide  which  forms  on 
the  surface  of  aluminum  conductors.  The  experience  of  many 
engineers  does  not,  however,  confirm  this.  Mr.  W.  T.  Taylor 
says  that  copper  collects  a  little  more  sleet  than  aluminum. 
He  bases  this  statement  on  observations  on  long-distance  tele- 
phone lines  in  California,  where  copper  and  aluminum  conductors 
run  side  by  side.  Mr.  E.  H.  Farrand1  has  observed  copper  wire 
accumulate  snow  to  the  extent  of  3  in.  in  diameter  in  a  few  min- 
utes, while  iron  wire  took  a  coating  ^  in.  to  1  in.  thick;  but  after 
a  short  time  the  wires  of  both  materials  had  accumulated  snow  to 
the  same  diameter.  Messrs.  R.  B.  Matthews  and  C.  T.  Wilkin- 
son2 have  made  many  observations  in  the  sleet  districts  of  the 
United  States,  and  they  are  of  opinion  that  an  aluminum  con- 
ductor will  collect  as  much  sleet  and  ice  as  copper  or  steel  wires. 

When  wires  are  hung  vertically  one  above  the  other,  in  the 
manner  frequently  adopted  in  the  earlier  constructions  for  double- 
circuit  lines,  there  is  the  possibility  of  sleet  falling  off  one  of  the 
lower  wires,  while  the  upper  wire  remains  heavily  loaded  and 
with  considerable  sag.  This  might  cause  the  lower  wire  to  rise 
into  contact  with  the  upper  wire.  With  this  possibility  in 
mind,  the  disposition  or  spacing  can  be  made  so  that  short- 
circuits  due  to  this  cause  are  not  likely  to  occur. 

Sleet  storms  are  not  infrequently  followed  by  low  tempera- 
tures and  high  winds.  The  loading — especially  when  the 
conductors  are  of  small  diameter — is  then  liable  to  be  so  great 
that  it  may  be  false  economy  to  guard  against  it  by  increasing 
the  factor  of  safety.  It  is  probable  that  means  will  be  adopted  in 
the  future  to  prevent  the  formation  of  sleet  deposits  on  overhead 
power  conductors  by  passing  sufficient  current  through  them  to 
raise  the  temperature  above  that  at  which  the  deposit  will  form. 
On  many  systems  it  is  possible  to  increase  the  current  in  the  lines 
by  controlling  the  power  factor  of  the  load.  Users  of  power 
might  assist  in  maintaining  service  during  severe  sleet  storms  by 
under  exciting  synchronous  motors  connected  to  the  transmission 

1  Journal  of  the  Inst.  E.  E.,  p.  659,  vol.  46  (1911). 

2  "Extra  High  Pressure  Transmission  Lines."     Jour,  Inst.  E.  E.,  p.  573, 
vol.  46  (1911). 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  267 

system.  If  sufficient  current  cannot  be  obtained  by  such  means, 
the  procedure  on  systems  where  there  is  a  duplicate  line  would 
be  to  throw  all  the  load  on  one  line  and  pass  a  suitable  current 
through  the  spare  line  which  would  be  disconnected  from  the 
load  and  short-circuited  at  the  receiving  end. 

The  watts  per  square  inch  of  cooling  surface  necessary  to  raise 
the  temperature  sufficiently  to  prevent  sleet  deposits  will  depend 
largely  upon  whether  or  not  there  is  much  wind  at  the  time  of  the 
sleet  storm;  but  something  under  34  watt  per  square  inch  should 
suffice.  This  is  a  problem  in  connection  with  which  data  obtained 
from  various  districts  where  sleet  deposits  are  common  would  be 
valuable. 

Wind  Pressures. — The  pressures  due  to  winds  of  high  velocity 
acting  on  poles  and  wires  in  a  direction  approximately  at  right 
angles  to  the  transmission  line  are  of  great  importance,  both 
where  sleet  formation  is  possible,  and  in  districts  where  sleet 
cannot  form.  In  the  latter  case,  the  velocity  of  the  wind  is 
frequently  greater  than  in  the  colder  districts;  but,  on  the  other 
hand,  a  moderate  wind  acting  on  the  larger  diameter  of  ice- 
coated  wires  will  generally  lead  to  the  greatest  stressing  of 
the  conductor  material.  The  maximum  wind  velocities  rarely 
occur  at  the  lowest  temperatures,  but  falling  temperatures  and 
rising  wind  are  not  unusual  after  sleet  storms.  In  mountainous 
districts  a  transmission  line  may  be  subjected  at  certain  points 
to  gusts  of  wind  blowing  almost  vertically  downward;  the  pres- 
sure in  such  a  case,  being  directly  added  to  the  weight  of  wire 
and  the  ice  load,  may  lead  to  more  serious  results  than  an  even 
stronger  wind  blowing  horizontally  across  the  line. 

Numerous  observations  have  been  made  on  wind  pressures, 
and  it  is  found  that  the  pressure  exerted  on  small  surfaces  is 
proportional  to  the  square  of  the  wind  velocity.  It  will  also 
depend  to  some  small  extent  upon  the  density  of  the  air,  or 
the  barometric  pressure;  but  the  correction  for  barometric 
pressure  is  usually  not  worth  making. 

Wind  Velocity. — It  is  well  to  distinguish  between  indicated 
and  true  wind  velocities.  The  United  States  Weather  Bureau 
observations  are  made  with  the  cup  anemometer,  and  wind 
velocities  over  short  periods  of  time  are  calculated  on  the  as- 
sumption that  the  velocity  of  the  cups  is  one-third  of  the  true 
velocity  of  the  wind,  for  great  and  small  velocities  alike.  This 
assumption  is  not  justifiable,  and  a  correction  must  therefore  be 


268 


ELECTRIC  POWER  TRANSMISSION 


made  in  order  to  convert  the  Weather  Bureau  recorded  velocities 
into  true  velocities.  The  actual  wind  velocities  corresponding 
to  definite  indicated  velocities,  as  given  by  the  U.  S.  Weather 
Reports,  are  as  follows: 

WIND  VELOCITY:  MILES  PER  HOUR 


Indicated 


Actual 


10                 ..... 

9  6 

20      .  .     

17  8 

30  
40    

25.7 
33  3 

50  

40.8 

60 

48  0 

70 

55  2 

80 

62  2 

90      '       ...     .      ... 

69  2 

100  

76.2 

Unless  otherwise  stated,   when   wind   velocity  is    referred   to, 
this  must  be  understood  to  be  the  true  velocity. 

Formulas  for  Wind  Pressures. — The  formula  proposed  by  the 
U.  S.  Weather  Bureau  (Professor  C.  F.  Marvin),  giving  pressure 
in  pounds  per  square  foot  on  small  flat  surfaces  normal  to  the 
direction  of  the  wind  is: 

D-  V2  (126) 


where  B  is  the  barometric  reading  in  inches,  and  V  is  the  wind 

velocity  in  miles  per  hour.     Other  formulas  are: 

Langley F  =  0.0036F2 

Smeaton...  ..  \F=0-005F, 

In  the  case  of  cylindrical  wires,  the  pressure  per  square  foot 
of  projected  area  is  less  than  on  flat  surfaces.  The  formula 
proposed  by  Mr.  H.  W.  Buck,1  and  generally  conceded  to  be 
correct  is: 

F  =  0.0025Y2  (127) 

where  F  is  the  pressure  per  square  foot  of  projected  surface  of  a 
cylindrical  wire.  A  more  convenient  form  of  expression  for  this 
relation  is: 

dV2 


P      5000 
In  paper  read  at  the  World's  Fair,  St.  Louis,  1904. 


(128) 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  269 

where  p  is  the  pressure  per  foot  length  of  the  wire  and  d  is  the 
diameter  of  the  wire  in  inches;  and  the  denominator,  although 
not  exactly  as  obtained  from  Mr.  Buck's  equation,  is  an  easily 
remembered  round  number,  which  is  very  close  to  the  average 
of  many  experimental  results.  The  upper  curve  in  Fig.  96 
has  been  plotted  from  Mr.  Buck's  formula  for  cylindrical  wires; 
while  the  lower  curve  for  pressures  on  flat  surfaces  gives  the  rela- 
tion between  pressure  and  wind  velocity  according  to  Professor 
Langley's  formula. 


0  5  101520258085404660 

Pressure- Pounds  per  Square  Foot 

FIG.  96. — Curves  giving  relation  between  wind  velocity  and  resulting  pressure. 


Relation  between  Wind  Velocity  and  Height  above  Ground. — 
Owing  to  the  resistance  offered  by  the  ground  surface,  the  force 
of  the  wind  is  not  so  great  near  the  ground  as  at  higher  altitudes, 
and  greater  maximum  wind  pressures  on  wires  must  be  allowed 
for  when  the  line  is  carried  on  high  steel  structures  than  in  the 
case  of  the  average  wood  pole  line,  as  used  for  the  lower  voltage 
transmissions. 

Mr.  F.  F.  Fowle1  has  given  valuable  particulars  based  on 
maximum  wind  velocities  at  different  elevations,  observed  in 
Chicago,  from  which  the  curve  of  Fig.  97  has  been  plotted.  It 
will  be  seen  that,  at  the  average  elevation  of  transmission  line 
wires  (25  to  45  ft.),  the  probable  maximum  wind  velocity  is  less 

1  "A  Study  of  Sleet  Loads  and  Wind  Velocities."  Electrical  World,  New 
York,  October  27,  1910. 


270 


ELECTRIC  POWER  TRANSMISSION 


than  half  what  it  would  be  at  100  ft.  above  ground,  and  only 
about  one-third  of  what  may  be  expected  at  an  elevation  of  300  ft. 
Mr.  Fowle  suggests  that  overhead  line  calculations  might  be 
based  on  a  maximum  wind  velocity  of  47  miles  per  hour  for 
ordinary  steel  tower  construction,  and  40  miles  for  wood  pole 
lines.  These  are  probably  safe  limits,  especially  in  climates 
where  this  maximum  wind  pressure  is  considered  as  acting  on 
ice-coated  wires.  In  exposed  positions,  and  where  the  line  runs 


Maximum  Wind  Velocity-  Miles  per  Hour 
S  8  g  S  g  8  3  S  8  ! 

/ 

/  - 

/ 

- 

/ 

- 

/ 

- 

/ 

/ 

- 

/ 

- 

[/ 

• 

- 

- 

50  100  150          -w          ~v          ow 

Height  above  Ground  Level  (Feet) 

FIG.  97. — Maximum  wind  velocities  at  different  elevations. 

through  wide  stretches  of  open  country,  it  is  well  to  allow  a 
maximum  of  60  for  steel  tower  lines,  and  50  for  wood  pole  lines. 

According  to  Mr.  Fowle,  the  wind  velocity  very  rarely  exceeds 
50  miles  in  Chicago,  the  maximum  recorded  during  a  period  of 
thirty-six  years  being  84,  while  90  miles  per  hour  has  been  re- 
corded in  Buffalo.  Wind  velocities  of  53,  and  on  one  occasion 
60  miles,  have  been  recorded  during  sleet  storms;  but  such 
velocities  are  exceptionally  high. 

The  revised  British  Board  of  Trade  requirements  for  the  cal- 
culation of  pole  lines  are  a  limit  of  25  Ib.  per  square  foot  on  flat 
surfaces,  and  25  X  0.6  =  15  Ib.  on  the  projected  surface  of 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  271 

cylindrical  poles  and  wires.  No  allowance  for  ice  coating  is  made ; 
but  since  it  is  specifically  stated  that  no  allowance  is  to  be  made 
for  the  elasticity  of  the  wires,  and  the  factors  of  safety  insisted 
upon  are  5  for  wires,  6  for  steel  poles,  and  10  for  wood  poles, 
there  is  very  little  danger  of  electric  power  lines  being  damaged 
by  storms  in  England.  Whether  or  not  such  stringent  regula- 
tions are  necessary,  or  likely  to  encourage  the  development  of 
power  transmission  schemes,  is  another  question,  the  answer  to 
which  is  generally  known. 

Calculations  based  on  British  B.  0.  T.  requirements  are  made 
for  a  temperature  of  22°  F.  At  this  temperature,  the  sag  of  a  No. 
2/0  copper  conductor  on  a  600-ft.  span  would  be  24.6  ft.  to  satisfy 
the  requirements.  At  122°  F.  (the  approximate  maximum  sum- 
mer sun  temperature  in  England)  the  sag  of  the  same  wire  would 
be  33  feet.  With  aluminum  conductors  the  conditions  would 
be  worse. 

In  America,  where  the  weather  conditions  are  more  severe,  the 
larger  and  more  economical  spans  are  very  common;  but 
there  are  few  interruptions  to  service  due  to  broken  wires  not- 
withstanding the  smaller  factors  of  safety  as  usually  employed 
on  this  side  of  the  Atlantic.  The  fact  that  an  overhead  trans- 
mission line  is  a  very  flexible  and  elastic  structure  appears  fre- 
quently to  be  overlooked  by  the  inventors  whose  genius  finds  an 
outlet  in  the  framing  of  rules  and  regulations. 

The  Committee  on  Overhead  Line  Construction,  appointed  by 
the  National  Electric  Light  Association  of  New  York,  assumes  a 
half-inch  ice  coating  for  all  sizes  of  conductors,  and  maximum 
wind  velocities  of  50  to  60  miles  per  hour.  This  committee  states 
that  62  miles  is  a  velocity  not  likely  to  be  exceeded  during  the 
cold  months. 

Three  classes  of  loading  are  considered  by  the  Joint  Committee 
on  Overhead  Crossings,  particulars  of  which  are  as  follows: 


Class  of  loading 

Vertical  component 

Horizontal  component, 
Ib.  per  sq.  ft.  of 
projected  area 

A 

dead 

15 

B 

dead  +  M-in  ice 

s 

C  

dead  +  %-in.  ice 

11 

For  the  class  B  loading  the  ordinary  range  of  temperature  is 
given  as  —20  to  120°  F.     For  the  calculation  of  pressures  on 


272  ELECTRIC  POWER  TRANSMISSION 

supporting  structures  the  requirements  are  13  Ib.  per  square  foot 
on  the  projected  area  of  closed  or  solid  structures,  or  on  1% 
times  the  projected  area  of  latticed  structures.  The  same  Joint 
Committee  allows  a  safe  stress  on  copper  50  per  cent,  of  ultimate 
breaking  stress — that  is  to  say,  the  wires  may  be  stressed  to  a 
point  very  near  to  the  elastic  limit. 

The  proposed  National  Standards,  as  formulated  by  the  Bureau 
of  Standards,  U.  S.  Dept.  of  Commerce1  cover  three  grades  of 
overhead  construction: 

(a)  power  lines  crossing  railways; 

(b)  power  lines  crossing  unimportant  railways; 

(c)  power  lines  in  country  districts  where  the  risk  of 
accident  from  falling  wires  is  small. 

Three  classes  of  loading  are  also  considered: 

(1)  Heavy  loading:  the  resultant,  at  a  temperature  of  0°  F., 
due  to   dead   weight  of  wire  plus  J^  in.  radial  thickness  of  ice, 
with  horizontal  wind  pressure  of  8  Ib.  per  sq.  ft.  of  the  projected 
surface  of  the  ice-coated  wire. 

(2)  Medium   loading:   at   a   temperature  of  15°  F.,  a  total 
load  equal  to  %  of  (1)  with  a  minimum  of  1.25  times  the  dead 
weight  of  the  conductor  (without  ice  covering). 

(3)  Light  loading:  at  a  temperature  of  30°  F.,  a  total  load 
equal  to  %  of  (2)  or  %  of  (1)  with  a  minimum  of  1.25  times  the 
dead  weight  of  the  conductor  (without  ice  covering). 

The  Board  of  Railway  Commissioners  for  Canada  specifies  for 
H.  T.  wires  crossing  railways,  a  factor  of  safety  of  2  when  wires 
are  coated  with  ice  or  sleet  to  a  depth  of  1  in.,  and  subject  to 
wind  pressure  of  100  miles  per  hour.  This  combination  of  ab- 
normal loads  being  more  or  less  imaginary,  it  is  probable  that  the 
factor  of  safety  is  actually  about  6,  which  appears  to  be  un- 
necessarily high. 

A  more  reasonable  specification  for  railroad  crossings — al- 
though probably  unduly  severe,  and  therefore  leading  to  un- 
necessary capital  outlay — is  that  of  the  New  York  Central  & 
Hudson  River  Railroad,  which  provides  for  ^  in.  ice  coating 
with  a  wind  pressure  of  20  Ib.  per  sq.  ft.  of  projected  area,  and  a 
limiting  stress  equal  to  j>{  Q  of  the  ultimate  stress  of  the  wire. 

1  Taken  from  the  National  Electric  Safely  Code,  second  edition,  Nov.  15, 
1916. 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  273 

132.  Swaying  of  Wires  in  Strong  Winds. — If  a  transmission 
line  is  well  designed  and  constructed,  all  the  wires  of  one  span  will 
generally  be  found  to  swing  synchronously  in  any  wind.     Under 
exceptional  conditions,  however,  trouble  is  liable  to  occur  through 
wires  swinging  together,  even  when  all  details  of  design  and  con- 
struction  have   received    careful   attention.     Troubles   of   this 
description  are  more  likely  to  be  met  with  when  the  spans  are 
long   and  the  sag  in  the  wires  necessarily  large,  and  for  this 
reason  the  spacing  between  wires  must  increase  with  increase 
of  span  length,  irrespective  of  voltage  considerations.     Copper 
conductors  are  decidedly  less  likely  to  swing  out  of  synchronism 
than  aluminum  conductors;  not  only  because  the  latter  have 
usually  to  be  strung  with  a  greater  sag,  but  also  because  of  the 
lightness  of  the  material.     Aluminum  conductors  of  small  sec- 
tion are  easily  shaken  by  sudden  gusts  of  wind,  and  a  little 
difference  in  sag  will  in  all  probability  lead  to  non-synchronous 
swinging.     It  must  not  be  overlooked  that  wires,  after  erection, 
do  not  always  remain  equally  taut.     This  may  be  due  to  many 
causes,  such  as  a  slight  slipping  in  the  ties,  straining  of  insulator 
pins  on  cross-arms,  unequal  ice  loading,  or  local  faults  in  the 
wires  themselves.     Again,   it   has  been  observed   that   during 
snow  storms  all  the  wires  do  not  always  become  coated  to  an 
equal  extent,  and  such  a  want  of  uniformity  in  the  ice  coating 
may  well  lead  to  wires  being  blown  together  in  a  strong  wind. 

On  the  high-tension  transmission  system  of  the  Central 
Colorado  Power  Company,  with  spans  averaging  730  ft.,  the 
lines  cross  some  very  exposed  positions  at  the  openings  to  can- 
yons, and  the  excessively  strong  winds  that  occur  at  such  points 
have  been  known  to  mix  up  the  conductors.  It  was  found 
necessary  to  dead-end  the  line  at  each  tower,  guy  the  towers,  and 
increase  the  tension  in  the  wires  to  a  point  near  the  elastic  limit  of 
the  material,  steel  being  used  where  necessary  in  lieu  of  copper. 

133.  Calculation  of  Total  Stress  in  Overhead  Wires.— The 
formula  (101)  of  Article  125  gives  the  relation  between  tension, 
sag,  and  weight  for  a  wire  strung  between  supports  a  known 
distance  apart.     Thus,  if  the  tension,  P,  is  known  or  assumed, 
the  sag  can  readily  be  calculated.     We  shall  now  consider  how 
this  tension  can  be  computed,  not  only  when  the  wire  carries  an 
increased  vertical  load  in  the  form  of  ice  deposit;  but  also  when 
the  effect  of  wind  blowing  across  the  line  increases  the  total  stress. 

If  wr  is  the  total  loading  in  Ib.  per  foot  run  of  the  wire,  and  w 


274 


ELECTRIC  POWER  TRANSMISSION 


FIG.  98. — Vector  diagram  of 
forces  acting  on  overhead 
wires. 


is  the  weight  in  Ib.  per  foot  length  of  the  unloaded  wire,  then 
wr  =  nw,  where  n  is  a  multiplier  which  takes  account  of  the 
extra  load  on  the  wire  under  the  most  severe  weather  conditions 
likely  to  be  encountered  in  the  district  where  the  transmission 
line  is  erected. 

It  is  usual  to  assume  that  the  wind  pressure  acts  in  a  horizontal 
direction  and  that  the  total  load  on  a  conductor  is  the  resultant 
of  two  forces,  one  acting  vertically 
downward  due  to  weight  of  wire 
together  with  added  weight  of  sleet 
or  ice,  if  any,  and  one  acting  hori- 
zontally due  to  the  wind  pressure. 
These  forces  are  indicated  in  Fig.  98 
where  Op  represents  the  wind  pressure, 
Ow  the  weight  of  the  conductor,  and 
ww\  the  added  weight  of  ice.  The 
resultant  pressure  Owr  is  equal  to 
Vp2  +  Wi2.  If  the  line  runs  through 
a  country  where  sleet  does  not  form 
on  the  wires  the  maximum  resultant 
pressure  is  Or  instead  of  Owr  if  the  assumed  maximum  force  due 
to  wind  is  the  same  in  both  cases. 

The  diagram  Fig.  99  gives  values  of  the  multiplier  n  (i.e.,  the 
ratio  Or  -r-  Ow  of  Fig.  98)  corresponding  to  various  wind  veloci- 
ties for  standard  sizes  of  solid  copper  conductors  on  the  assump- 
tion that  there  can  be  no  ice  formation  on  the  wires,  while  Fig. 
100  gives  values  of  n  (i.e.,  the  ratio  Owr  +  Ow)  for  copper  con- 
ductors when  the  weight  is  increased  by  a  coating  of  ice  0.5  in. 
thick  with  a  correspondingly  greater  wind  effect  due  to  the  in- 
creased diameter.  The  curves  of  Figs.  101  and  102  give  similar 
relations  but  for  conductors  of  aluminum  instead  of  copper. 

The  formula  used  for  the  calculation  of  wind  pressure  in  con- 
nection with  these  diagrams  is 

p  =  dV*  -5-  4820 

where  d  is  the  diameter  in  inches  of  the  conductor  or  of  the  ice 
coating,  as  the  case  may  be;  V  is  the  actual  wind  velocity  in 
miles  per  hour,  and  p  is  the  wind  pressure  in  pounds  per  foot 
length  of  conductor. 

This  is  the  more  correct  form  of  the  formula  (128)  already 
given.  When  using  the  diagrams,  it  should  be  noted  that  the 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  275 

distances  plotted  horizontally  represent  the  squares  of  the  wind 
velocities,  and  the  sizes  of  the  conductors  are  expressed  in  equiva- 
lent B.  &  S.  gauge  numbers  or  in  circular  mils  for  the  larger  sizes. 


wajonpnoo  J9ddoo  pjjog  jo  ja^aajcja 


The  curves  of  Fig.  99  are  correct  for  solid  copper  wires:  the 
values  of  n  for  stranded  conductors  would  be  somewhat  greater 
because  of  the  larger  surface  presented  to  the  wind  for  the  same 


276 


ELECTRIC  POWER  TRANSMISSION 


2        2       3        3        3-       2 

jojonpuoo  ranajraniv  popxrea^g  P  »»9«reia 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  277 

vertical  loading.  The  error  introduced  by  using  Fig.  99  for 
stranded  cables  is  not  of  great  practical  importance. 

The  curves  of  Fig.  101  are  approximately  correct  for  stranded 
aluminum  conductors;  but  since  the  actual  diameter  will  vary 
with  the  method  of  stranding,  these  charts  are  intended  only  for 
the  use  of  practical  engineers  who  are  not  interested  in  mathe- 
matical niceties.  The  calculations,  on  the  basis  of  the  assump- 
tions previously  made,  are,  however,  very  simple.  Thus,  if 
6  stands  for  the  angle  wOr  or  w\OwT  as  the  case  may  be  (see  Fig. 
98),  we  may  write: 

horizontal  loading 

tan  6  = 7^ — ; 

vertical  loading 

whence  9,  and  therefore  sec  6  can  be  obtained  from  trigonomet- 

.    resultant  loading 
rical  tables.     This  last  quantity,  being  the  ratio  vertical  ioadillg 

is  the  required  factor  n  when  ice  loading  is  not  considered. 
The  correction  to  be  made  when  the  vertical  load  includes  ice 
deposit  is  simple  and  obvious. 

The  special  case  of  solid  wires  without  ice  coating  can  be 
treated  as  follows: 

dV2 
Wind  pressure  per  foot  length p  =  ^^Q 

Vertical  load  per  foot  length  (copper) w  =  3.02d2 

Vertical  load  per  foot  length  (aluminum),  w  =  Q.92dz 


(129) 

'W~ 

whence,  for  solid  wires  without  ice  deposit, 


n  (for  copper)  =  ^    1  +  d2  x  21         x  1Q4  (130) 


I  V* 

n  (for  aluminum)  =  ^  1  +  rf2  x  1Q65  x  1Q4         (131) 

Example:  What  is  the  ratio  of  the  resultant  load  with  actual 
wind  velocity  of  70  miles  per  hour,  to  normal  load  with  wire 
hanging  in  still  air,  in  the  case  of  a  No.  6  B.  &  S.  copper 
conductor? 

The  diameter  is  d  =  0.162  in.,  and  by  formula  (130) 


(0.162)2  X  21,200  X  104 


278  ELECTRIC  POWER  TRANSMISSION 

134.  Effect  of  Temperature  Variations  on  Sag  and  Stress.  — 
Assuming  the  tension  at  all  parts  of  the  wire  to  be  the  same,  and 
equal  to  P  lb.,  the  formula  (102)  of  Article  125  may  be  written: 


where  I  and  s  are  in  feet,  and  w  is  the  weight  per  foot  of  the  wire. 
If  wind  or  ice  or  both  result  in  a  loading  per  lineal  foot  equal  to 
to  nw,  it  follows  that  the  tension  in  the  loaded  wire  is  now  n 
times  as  great.  If  the  symbol  S  be  used  to  denote  the  stress 
or  tension  in  the  wire  per  square  inch  of  cross-section,  the  formula 
giving  the  relation  between  tension  and  sag  may  be  written, 

S  =  kn-  (133) 

$ 

in  which  k  is  constant  for  a  given  material;  it  is  equal  to  g-r  where 

A  is  the  cross-section  of  the  conductor  in  square  inches.  The 
numerical  value  of  k  is  therefore  one-eighth  of  the  weight  in 
pounds  of  12  cu.  in.  of  conductor  material,  or  1.5  times  the  weight 
of  a  cubic  inch.  The  values  of  k,  together  with  ultimate  and 
working  values  of  the  stress  S,  will  be  found  in  the  table  of  physical 
constants  in  Article  41  of  Chapter  IV  (p.  75).  The  length  of  wire 
between  fixed  supports  of  equal  height  bears  a  definite  relation 
to  the  span  and  sag.  This  relation,  as  already  given,  is: 

x  -  i  +'|!  (in) 

where  I  is  the  distance  between  supports  and  s  the  sag  at  center, 
both  expressed  in  feet. 

The  increase  in  length  due  to  stress  will  be  directly  propor- 
tional to  the  tension  per  square  inch  (S)  provided  the  elastic  limit 
of  the  material  is  not  exceeded.  The  approximate  elastic  limit 
for  conductor  materials  is  given  in  the  table  of  Chapter  IV. 
The  formula  for  elastic  stretching,  as  already  given,  is 

Xe  =  X^  (125) 

where  X  is  the  length  of  wire  when  S  =  0,  Xe  is  the  elongation  due 
to  the  stress  S,  and  M  is  the  elastic  modulus. 

The  increase  of  length  due  to  rise  of  temperature,  on  the  as- 
sumption that  stress  remains  unaltered,  is: 

X  X  a  X  i  (134) 

where  a  is  the  temperature-elongation  coefficient. 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    279 

Values  of  k,  a,  M  and  the  maximum  safe  working  stress  S, 
taken  from  the  data  in  Chapter  IV,  are  as  follows: 

For  copper k  =  0.485;  a  =  0.0000096; 

M  =  15,000,000;  S  =  28,000 

For  aluminum k  =  0.146;  a  =  0.0000128; 

M  =  9,000,000;  S  =  13,000 
For  copper-clad  steel  (40  per  cent.) k  =  0.447;  a  =  0.0000067; 

M  =  20,000,000;  S  =  40,000 
For  iron  wire k  =  0.423;  a  =  0.0000066; 

M  =  25,000,000;  S  =  26,000 
For  Siemens-Martin  steel k  =  0.427;  a  =  0.0000066; 

M  =  29,000,000;  S  =  33,000 

It  is  a  simple  matter  to  calculate  by  means  of  formula  (132) 
or  the  modified  formula  (133)  the  sag  corresponding  to  any 
span  (0,  load  (nw)  and  tension  (S),  and  by  using  either  of 
these  formulas,  the  minimum  allowable  sag,  for  the  safe  limiting 
tension  when  the  wire  is  subject  to  the  greatest  expected  load 
in  the  matter  of  ice  and  wind  pressure,  should  be  determined  in 
the  first  instance.  Having  determined  the  amount  of  this  sag — 
which  should  be  that  corresponding  to  the  lowest  expected  tem- 
perature— the  length  of  the  wire  in  the  span  can  readily  be  cal- 
culated by  means  of  formula  (111). 

The  calculation  of  the  sags  and  corresponding  tensions  at  other 
temperatures  and  under  other  conditions  of  loading  is  not  by 
any  means  so  simple  a  matter,  because  the  alteration  in  the 
length  of  the  conductor  depends  not  only  upon  the  temperature, 
but  also  upon  the  tension.  If  the  extra  load  on  the  conductor 
due  to  wind  pressure  and  ice  (if  any)  be  removed,  the  sag  will 
adjust  itself  until  the  formula  (132)  is  again  satisfied,  the  weight 
w  being  in  this  case  that  of  the  wire  only.  A  further  condition 
is  that  the  length  of  wire  shall  be  equal  to  the  length  as  origi- 
nally calculated  for  the  loaded  wire,  less  the  elastic  contraction 
due  to  the  reduction  in  the  tension.  If  the  temperature  be  now 
supposed  to  rise,  the  length  of  the  wire  will  increase,  but  not  in  di- 
rect proportion  to  the  temperature  rise  as  indicated  by  formula 
(134)  because  so  soon  as  there  is  any  increase  in  length  leading  to 
an  increased  sag,  the  tension  in  the  wire  is  immediately  relieved, 
and  since  it  is  assumed  that  the  elastic  limit  has  not  been  ex- 
ceeded there  will  be  a  reduction  in  length  which  could  be  cal- 
culated by  formula  (125)  if  the  amount  by  which  the  tension  is  re- 
lieved were  known. 


280  ELECTRIC  POWER  TRANSMISSION 

A  mathematical  formula  which  expresses  the  required  length 
or  sag  under  normal  conditions,  in  terms  of  the  length  correspond- 
ing to  minimum  temperature  and  heaviest  loading,  is  very  com- 
plex and  difficult  of  solution.  Mr.  H.  J.  Glaubitz1  has  evolved  an 
equation  in  which  the  first  and  third  power  of  the  unknown 
quantity  (the  deflection  or  sag)  appear  simultaneously.  The 
solution  of  such  an  equation  is  tedious  and  is  usually  accomplished 
with  the  assistance  of  more  or  less  scientific  guesswork.  A 
graphical  method,  which  is  probably  in  more  general  use,  consists 
in  plotting  two  curves,  one  showing  the  relation  between  sag  and 
tension  for  the  selected  span  when  the  wire  hangs  naturally  under 
its  own  weight  only,  and  another  curve  calculated  for  a  definite 
constant  temperature  and  giving  the  relation  between  sag  and 
tension  when  a  wire  of  definite  known  length  under  a  known 
tension  is  subjected  to  various  assumed  changes  in  the  tension. 
The  point  where  the  two  curves  cross  will  indicate  the  required 
conditions  of  sag  and  tension.  This  process  is  a  lengthy  and 
laborious  one  and  has  to  be  repeated  for  every  assumed  change  of 
temperature. 

Graphical  methods  of  calculating  sags  or  tensions  under 
various  conditions  of  temperature  and  load  are  quite  accurate 
enough  for  practical  purposes,  and  since  the  material  published 
in  this  connection  is  so  abundant  and  varied  that  there  is  room  for 
individual  choice  in  the  selection  of  a  particular  method,  or 
chart,  or  combination  of  charts  and  diagrams,  the  needs  of  the 
engineer  who  prefers  graphical  methods  to  the  more  lengthy 
analytical  processes,  are  easily  satisfied. 

For  one  who  is  working  constantly  on  the  same  kind  of  problem, 
diagrams  or  charts  are  usualy  of  very  great  assistance;  but  if  a 
method  involving  any  but  the  simplest  diagrams  is  not  made  use 

1  Electrical  World,  March  25,  1909,  p.  731.  The  reader  may  also  refer  to 
the  Electrical  World  of  July  13,  1912,  p.  101,  where  Mr.  H.  V.  Carpenter 
evolves  a  formula  containing  the  third  and  first  powers  of  the  unknown 
quantity,  and  proposes  a  chart  to  assist  in  arriving  at  the  solution.  An 
excellent  article  on  Sags  and  Tensions  in  wire  spans  from  the  pen  of  Dr. 
Harold  Fender  appeared  on  page  604  of  the  Electrical  World  of  Sept.  28, 
1907.  Among  more  recent  contributions  there  is  an  article  by  Mr.  K.  L. 
Wilkinson  in  the  Electrical  World  of  Feb.  6,  1915,  and  a  paper  by  Mr.  A.  T. 
Arnall  on  p.  360  of  vol.  cci  (June,  1916)  of  the  Proceedings  of  the  Institution 
of  Civil  Engineers  (England).  Practical  suggestions  for  solving  the  equa- 
tions containing  the  third  power  together  with  the  first  or  second  power  of 
the  unknown  quantity  are  made  in  these  papers. 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  281 

of  very  frequently,  mistakes  are  likely  to  occur,  or  else  the  time 
spent  in  re-studying  the  method  of  procedure  leaves  the  graph- 
ical treatment  without  advantage  over  the  more  tedious  process 
involving  the  solution  of  mathematical  equations. 

The  writer  has  always  found  charts  or  diagrams  of  great 
assistance  in  practical  engineering  work,  and  in  the  first  edition 
of  this  book  an  accurate  method  of  solving  sag-temperature 
problems  by  means  of  two  superimposed  diagrams  was  described; 
but,  for  the  reasons  above  given  and  also  to  avoid  the  introduc- 
tion of  unnecessary  material,  the  graphical  method  of  procedure 
is  omitted  from  this  edition.  The  analytical  method  about 
to  be  described  does  not  require  the  solution  of  equations  con- 
taining the  third  and  other  powers  of  the  unknown  quantity: 
it  is  accurate,  and  does  not  include  any  difficult  mathematical 
processes. 

135.  Calculation  of  Sags  and  Tensions  under  any  Conditions 
of  Load  and  Temperature. — This  method  of  calculation  requires 
the  knowledge  of  the  sag  and  tension  corresponding  to  one  par- 
ticular temperature.  A  different  sag  is  then  assumed,  and  the 
temperature  at  which  this  sag  will  occur  is  calculated  by  means  of 
a  simple  formula.  The  manner  of  obtaining  the  preliminary  data 
will  be  explained  later. 

It  is  assumed  that  the  conductor  is  strung  between  two  fixed 
supports  on  the  same  level,  and  that  the  material  of  the  conductor 
is  not  strained  beyond  the  elastic  limit. 

The  meaning  of  the  symbols  used  is  as  follows: 

I  =  the  length  of  span,  or  horizontal  distance  between 

points  of  support,  in  feet, 
s  =  the  vertical  sag  at  center  of  span,  in  feet,  when  wire 

hangs  in  still  air  under  the  influence  of  its  own  weight 

only. 
P  =  the  tension  in  the  conductor  at  the  lowest  point  of 

span,  in  pounds. 
S  =  the  stress  in  the  conductor  at  the  lowest  point  of  span, 

in  pounds  per  square  inch  of  cross-section. 
X  =  the  length  of  conductor  measured  between  the  two 

points  of  support. 

t  =  temperature,  in  degrees  Fahrenheit. 
sc,  Sc,  Xc  =  known  values  of    sag,  stress  and  length  cor- 
responding   to   a  definite  temperature  tc,  when  wire 


282  ELECTRIC  POWER  TRANSMISSION 

hangs  in  still  air  under  the  influence  of  its  own  weight 
only. 

a  =  the  coefficient  of  linear  expansion  of  the  conductor  per 

degree  Fahrenheit. 

M  =  the  modulus,  or  coefficient,  of  elasticity  of  the  con- 
ductor, being  the  ratio  of  stress  in  pounds  per  square 
inch  to  extension  per  unit  length. 

w  =  the  weight  of  conductor  in  pounds  per  foot  of  length. 
wr  =  the  resultant  or  total  load  in  pounds,  per  foot,  includ- 
ing wind  pressure  and  ice  (if  any). 

n  —  a  multiplier  depending  on  the  material  of  the  con- 
ductor and  weather  conditions,  being  the  ratio 
wr/w,  when  supports  are  on  the  same  level. 

k  =  a  constant  depending  upon  the  material  ofthe  con- 
ductor, being  1.5  times  the  weight  in  pounds  of  a 
cubic  inch  of  the  conductor  material. 

The  well-known  formula  giving  the  relation  between  sag, 
length  of  span,  horizontal  load,  and  tension  is: 

.  -  £  (101) 

The  approximate  formula  for  the  length  of  a  parabolic  curve 
(which  is  quite  sufficiently  accurate  for  practical  purposes)  is: 

x  =  J  +  |-2  (ill) 

It  is  assumed  that  the  sag  sc,  and  therefore  the  corresponding 
stress  Sc  and  length  Xc  are  known  for  the  particular  temperature 
lc,  which  may  be  fairly  high  so  that  another  value,  s,  of  the  sag, 
arbitrarily  chosen,  shall  be  smaller  than  sc]  and  it  is  proposed  to 
calculate  the  temperature  t  which  will  correspond  to  this  assumed 
sag,  s. 

With  the  reduction  in  the  amount  of  sag,  there  must  of  neces- 
sity be  a  reduction  in  the  length  Xc  of  the  conductor  and  an  in- 
crease in  the  tension  Pc  or  stress  Sc.  The  amount  by  which  the 
length  has  decreased  is  not  directly  proportional  to  the  reduc- 
tion in  temperature,  because  the  increase  in  tension  causes  an 
elastic  elongation  of  the  conductor,  and  the  reduction  in  length 
is  actually  the  difference  between  the  amount  by  which  the  wire 
would  contract  with  the  lowering  of  the  temperature  if  the  ten- 
sion were  to  remain  constant,  and  the  amount  by  which  the  wire 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  283 

would  be  extended,  due  to  the  increased  tension,  if  the  tempera- 
ture were  to  remain  constant;  although,  from  a  strictly  scientific 
point  of  view,  the  argument  may  be  inaccurate.  The  decrease 
in  length  due  to  temperature  reduction  is: 

Xc  X  a  X  (to  -  0  (135) 

and  the  increase  in  length  due  to  additional  tension  is: 

&(*%-«)•, 

M  \s  I  ' 


Therefore, 

Xc  -  X  =  Xc  X  a  X  (*c  -  0  -  jjScQ  -  l) 

The  lengths  X  and  Xc  can  be  eliminated  by  substituting  their 
values  in  terms  of  sag  and  length  of  span,  as  given  by  formula 
(111).  This  leads  to  the  equation: 

8sc*  -  8s2      ,_&/!«  _  i\  Q37) 

~  (3*2  +  8sc2)a  +  aMs 

This  formula  is  very  simple  to  use,  because  for  a  given  material 
and  size  of  wire,  it  may  be  written: 

tc-  1  =  ^-^  +  ^-1}  (138) 

A2  \S  / 

where  Ki,  K2,  and  K3  are  constants,  the  values  of  which  are: 

K!  =  8sc2  (139) 

Kz  =  (3Z2  +  8st2)a  (140) 


. 

Moreover,  since  8sc2  is  always  very  small  in  comparison  with 
3£2,  the  constant  K2  may,  for  nearly  all  practical  purposes,  be 
written  : 

K2  =  3al*    (approx.)  (142) 

This  value  of  K2  may  be  used  for  spans  up  to  500  ft.  if  the 
multiplier  n  does  not  exceed  12,  and  for  spans  up  to  1000  ft. 
if  n  does  not  exceed  6.  In  the  case  of  longer  spans,  in  which  the 
sag  is  relatively  large,  or  if  a  closer  approximation  is  required, 
the  more  exact  expression  (140)  should  be  used.  The  only 
unknown  quantities  in  equation  (137)  or  (138)  being  the  sag  s 


284  ELECTRIC  POWER  TRANSMISSION 

and  temperature  t,  it  follows  that,  by  inserting  any  numerical 
value  for  s  in  the  equation,  the  change  of  temperature,  and  there- 
fore the  actual  temperature  t  corresponding  to  the  assumed  value 
of  the  sag,  can  be  readily  calculated.  In  order  that  this  method 
of  calculation  may  be  of  practical  utility,  it  is  necessary  that  the 
sag  sc  and  the  stress  Sc  at  the  particular  temperature  tc  shall  be 
known.  The  fundamental  data  on  which  all  line  calculations  are 
based  must  include  the  limiting  or  maximum  allowable  value  of 
the  stress  and  the  conditions  of  maximum  loading  under  the  most 
severe  weather  conditions.  The  maximum  load  per  foot  being 
wr  and  the  weight  of  the  unloaded  wire  being  w,  it  follows  that 

the  ratio  —  =  n  will  be  greater  as  the  wind  conditions,  either 

without  ice  or  combined  with  a  coating  of  ice  on  the  wires,  are 
the  more  severe.  The  wires  will  generally  be  subject  to  the 
greatest  stress  at  times  when  strong  winds,  with  or  without  a  coat- 
ing of  ice  or  sleet,  occur  at  a  low  temperature,  because  the  low- 
ness  of  the  temperature  alone  will  account  for  a  considerable 
increase  in  the  tension. 

If  the  extra  load  on  the  wire  due  to  wind  and  ice  combined  is 
great  in  proportion  to  the  weight  of  the  wire,  the  maximum  de- 
flection will  usually  occur  under  winter  conditions;  but  there  will 
be  a  higher  temperature  at  which  the  sag  of  the  unloaded  wire 
hanging  in  still  air,  subject  to  its  own  weight  only,  will  be  exactly 
the  same  as  the  deflection  under  winter  conditions  when  subject 
to  wind  pressure  and  extra  load  of  ice  (if  the  line  runs  through  a 
district  where  sleet  and  ice  formation  is  possible).  This  tempera- 
ture, which  may  be  called  the  critical  temperature  for  the  ma- 
terial of  the  conductor  when  the  maximum  winter  loading  has 
been  determined,  is  easily  calculated;  and  its  numerical  value, 
together  with  the  known  value  of  the  sag  under  conditions  of 
maximum  load,  and  of  the  tension  corresponding  to  this  sag, 
may  be  used  in  equation  (137)  or  (138)  for  the  known  quantities 
tc,  sc  and  Sc. 

Calculation  of  Critical  Temperature  tc 

Let  Sm  be  the  stress  in  the  wire  under  the  most  severe  conditions 
of  load,  and  t0  the  temperature  at  which  this  stress  occurs.  The 
tension  Sc  will  be  equal  to  Sm  divided  by  n,  because,  at  the  crit- 
ical temperature  tc,  the  sag  is  the  same  as  the  maximum  deflection 
of  the  loaded  conductor,  but  the  weight  per  foot  of  length  is  in 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  285 

the  ratio  —  or  -•     (Refer  to  formula  (101),  bearing  in  mind  that, 

for  any  given  size  of  conductor,  the  stress  S  is  proportional  to 
the  tension  P.)  With  an  increase  of  temperature  from  t0  to  tc  the 
reduction  in  stress  is 


and  the  reduction  in  length  of  the  wire  due  to  this  difference  of 
tension  is 


It  is  required  to  calculate  the  temperature  rise  (te  —  to)  which 
will  produce  an  extension  exactly  equal  to  this  elastic  contraction, 
in  order  that  the  length  Xc  of  the  wire,  and  consequently  the  sag 
sc,  shall  remain  as  before. 

The  extension  due  to  temperature  rise  is 

Xc  X  a  X  (te  -  t0) 
and  the  required  equation  is : 

Xc  X  a  X  (tc  -  to)  =  ^  Sm  (l  -  i) 

or  ftJ-r'tl  -      Sm     d  . 

to)  ~MXaV 

The  curves  in  Fig.  103  give  the  relation  between  the  ratio  — 

(being  the  reciprocal  of  n)  and  the  temperature  rise  (tc  —  t0)  for 
stranded  cables  of  different  materials.  The  values  of  M  and  a, 
which  have  been  adopted  for  the  purpose  of  drawing  the  diagram, 
are: 

For  hard-drawn  copper,  M  =  15  X  106,  a  =  9.6  X  10~6 

For  hard-drawn  aluminum,      M  =    9  X  106,  a  =  1.28  X  1Q-5 
For  galvanized  steel,  M  =  25  X  106,  a  =  6.5  X  1Q-6 

Having  determined  the  critical  temperature  tc  at  which — it  is 
interesting  to  note — the  tension  in  the  wires,  if  correctly  strung, 
will  be  the  same  whatever  the  length  of  span,  the  sag  sc  can  be 
calculated  by  the  formula  (101),  or  by  the  more  convenient 
formula : 

kl2 


s.  (144) 


which  can  be  put  in  the  form 

knl* 


o  (145) 

Om 


286 


ELECTRIC  POWER  TRANSMISSION 


In  this  manner  the  numerical  values  of  the  quantities  tc,  sc,  and 
Sc  for  use  in  formula  (137)  or  (138),  are  obtained. 

Example. — Construct  a  sag-temperature  chart  for  the  use  of 
the  construction  engineers  in  the  field,  based  on  the  following  data : 

Horizontal  span  =  485  ft.  with  rigid  supports  on  the  same 
level. 


v 

\ 

Chart  f 
will  m 
hangin 
to  defl 
maxim 

Ordina 
Where 

iving  te 
ke  sag 
g  unloa 
ction  u 
urn  stre 

es  of  C 

P 

a  —  t 

M  -n 

mperature  rise  which 
Of  stranded  conductor 
fcd  in  still  air  equal 
nder  conditions  of 

\ 

\ 

N 

s 

\ 

< 

laxiraum  stress,  Ibs. 
er  sq.  inch 
^mp.coefficent 
odulus  of  elasticity 
itio  of  resultant  or 
tal  load  per  foot 
nder  worst  conditions 
)  weight  per  foot  of 
nloaded  wire 

\ 

\ 

s 

\ 

ss 

\ 

\ 

\ 

\ 

t 
n 

t 

u 

\ 

S 

\k 

\ 

s 

\ 

\ 

>  

k 

N 

\ 

\ 

X 

\ 

V 

% 

^    \ 

X..N 

\ 

i 

N%       ^ 

^s. 

\ 

XN 

<'** 

5 

\ 

< 

^s^ 

*V  fe 

\      S 

C 

%s 

K\ 

V 

^ 

$s^ 

is 

\N 

SN 

^s^ 

s\ 

\ 

^ 

^v"N 

^s 

k 

^^0\ 

^S 

i^ 

0.2       0.3       0.4        0.5        0.6       0.7        0.8        0.9       1.0 
Ratio  ^-(beins  reciprocal  of  multiplier  n  ) 

Fio.   103. — Chart  for  determining  "critical  temperature." 

Conductors:  stranded  aluminum  No.  2/0  B.  &  S. 

Stress  not  to  exceed  Sm  =  14,000  Ib.  per  sq.  in.  with  a  combined 
load  of  0.5  in.  ice  coating  and  47-mile  wind  at  a  temperature  t0  = 
-  20°  F. 

From   Fig.    102,  the  value  of  n  for  V2  =  (47)2  =  2210  is  8, 


whence 


I  =  °-125- 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  287 


From  Fig.  103  (tc  -  to)  =  106 
whence  te  =  106  -  20  =  86°  F. 

this  being  the  "critical  temperature"  at  which  the  vertical  sag 
of  the  unloaded  conductor  hanging  in  still  air  will  be  the  same  as 
the  maximum  deflection  of  the  loaded  conductor  under  the  speci- 
fied conditions. 


Other  required  values  are  Sc 

,    ,,..,.     „ 
By  formula  (144),   Sc 


=  1750,  and  k  =  0.146. 


0.146  X  (485)2  . 

~I750~ 


By  formula  (139),  Ki  = 
By  formula  (142),  K2  = 

By  formula  (141),  K3  = 


8  X  (19.6)2  =  3070 
3  X  485  X  485  X  1.28  X  10~8 
1750  X  105 


9  X  106  X  1.28 


By  formula  (138),  (86  -  t)  = 


3070  -  8s2 


15.2 


+  15.2 


19.6 


Temperature,  Degrees  Fahrenheit 

?sosgggggg§si 

/ 

180 

Size  Aluminum  No.  2-0 
Span  =485  Feet 

p 

/ 

/ 

/ 

/ 

190 

/ 

/ 

195 

4 

>* 

/ 

Note;  Figures  on  Curve 
Indicate  Required 
Tension  on 
Dynamometer 

/ 

'20, 

xl 

10 

/ 

/ 

215 

17           1  V2         18          18H          19          19^2         20 

Sag  at  Center  of  Span    (Feet) 
FIG.  104. — Sag-temperature  curve  for  use  when  stringing  wires. 

By  choosing  values  for  s  not  very  different  from  sc  the  corre- 
sponding temperatures  are  easily  calculated.     Thus 

when  s  =  16^  ft.,  t  =  -  15.8°  F. 
when  s  =  17M  ft.,  t  =  +  15.4°  F. 
when  s  =  18M  ft.,  t  =  +  48.4°  F. 
when  s  =  203^  ft.,  t  =  +  120°  F. 

With  the  aid  of  these  figures  a  curve  such  as  Fig.  104  is  readily 
plotted.     This  curve  gives  the  men  in  the  field  all  necessary 


288  ELECTRIC  POWER  TRANSMISSION 

information  for  the  correct  stringing  of  the  conductors,  whatever 
may  be  the  temperature  when  the  work  is  carried  out. 

136.  Tensions  in  Conductors  when  Spans  are  of  Different 
Lengths. — It  is  well  to  keep  the  consecutive  spans  in  a  trans- 
mission line  as  nearly  as  possible  of  the  same  length,  because, 
although  it  is  possible  to  string  the  wires  so  that  the  tension  shall 
be  the  same  in  all  spans  at  the  time  of  stringing  or  under  specified 
conditions  of  load  and  temperature,  there  will  be  an  unbalancing 
of  the  tensions  in  adjoining  spans  with  every  change  of  tempera- 
ture.    If  properly  strung,  the  wires  in  long  and  short  spans  should 
be  subjected  to  the  same  maximum  tension  under  the  severest 
conditions  of  loading,  and  the  condition  of  equal  tensions  will 
repeat  itself  at  the  higher  temperature — previously  referred  to 
as  the  critical  temperature — when  the  sag  of  the  unloaded  con- 
ductor is  the  same  as  the  deflection  of  the  loaded  conductor  at  the 
lower  temperature;  but,  except  when  the  deflection  at  center  of 
span  remains  unaltered,1  the  pull  on  each  side  of  a  supporting 
insulator  will  be  unbalanced.     It  is  partly  for  this  reason  that 
extra  long  spans  are  usually  "dead-ended"  on  guyed  poles  or 
strain  towers.    When  calculating  sags  in  spans  of  different  lengths, 
it  is  therefore  not  correct  to  assume  that  the  sag  is  always  directly 
proportional  to  the  square  of  the  length  of  span;  because  when  the 
wire  hangs  in  still  air  subject  to  its  own  weight  only,  this  propor- 
tionality exists  for  no  other  temperature  but  the  critical  tempera- 
ture as  determined  for  use  in  the  sag-temperature  calculations. 

137.  Tension  in  Different  Sized  Wires  on  the  Same  Span.— It 
may  be  questioned  whether,  having  calculated  the  sag- tempera- 
ture conditions  for  a  conductor  of  diameter  d\,  there  is  not  a  short 
cut  by  which  similar  relations  can  be  arrived  at  for  a  wire  of 
diameter  dz  when  the  length  of  the  span,  I,  remains  unaltered. 
There  does  not  appear  to  be  a  quick  way  of  obtaining  the  required 
results;  but  there  is  one  condition  that  holds: 


irThe  condition  is  that  the  quantity  —          —  shall  remain  constant. 

n  —  ni 

This  expression  is  derived  in  the  same  manner  as  the  formula  (143) :  there 
must  obviously  be  some  particular  value  of  the  wind  or  ice  loading  (corre- 
sponding to  the  factor  n\)  which,  in  conjunction  with  a  rise  of  temperature 
(t\  —  to)  will  cause  the  deflection  to  be  the  same  as  when  the  temperature  is 
t0  and  the  (maximum)  loading  is  n  times  that  due  to  the  weight  of  the  wire 
acting  alone.  The  necessary  relation  between  t\  and  «i  is  given  by  the  above 
formula. 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS    289 

Let  n  be  the  load  coefficient  (  —  }  for  conductor  di,  and  n2  the 

load  coefficient  for  conductor  dz  then,  since  I2  is  assumed  constant, 
it  follows  from  the  formula  (145)  that 


and  s*  =  s'xrT* 

which  is  true  only  when  the  stress  (£)  per  square  inch  of  cross- 
section  is  the  same  for  both  sizes  of  conductor  (the  material  of  the 
conductors  being  assumed  the  same). 

138.  Further  Example  Illustrating  Temperature-sag  Calcu- 
lations. —  Although  the  problem  may  not  be  of  great  practical 
utility,  it  will  serve  as  a  further  illustration  of  the  methods  of 
calculation  explained  in  Article  135. 

Determine  the  reduction  of  temperature  which  mil  make  the 
tension  in  the  unloaded  conductor  the  same  as  in  the  loaded  conductor 
when  the  resultant  load  is  n  times  the  weight  of  the  wire. 

When  calculating  the  "critical  temperature,"  the  formula 
(143)  was  developed.  This  enables  us  to  calculate  the  rise  of 
temperature  necessary  to  make  the  length  of  the  unloaded  con- 
ductor hanging  in  still  air  equal  to  that  of  the  same  conductor  at 
a  lower  temperature,  but  with  an  extra  load  due  to  wind  or 
ice,  or  to  both  combined.  The  problem  now  before  us  is  con- 
cerned with  the  stress,  which  must  be  the  same  for  the  loaded 
and  unloaded  conditions.  Since  the  stress  remains  unaltered, 
the  change  in  length  of  the  conductor  corresponding  to  the 
(necessarily)  smaller  sag  is  caused  by  reduction  of  temperature 
and  not  by  elastic  contraction. 

If  s    =  maximum    deflection    under    loaded    condition,    and 
Si  =  sag  of  the  unloaded  wire  when  the  temperature  has  fallen 
t°  F.  and  the  stress  S  is  again  equal  to  what  it  was  under  the 
loaded  condition,  then, 

s 

Sl=n 
#• 
S 
The  reduction  in  length  is, 

\!  X  a  X  t  =  \  -  \i 


19 


290  ELECTRIC  POWER  TRANSMISSION 

which  resolves  itself  into 

8«'(n«  -  1) 
~  3n2  X  a  X  I 

No  appreciable  error  will  be  introduced,  when  dealing  with 
spans  of  moderate  length,  if  the  length  of  wire  Xi  is  replaced  by  the 
length  of  span  I.  Hence, 

=    8s*(n*  -  1) 
3n2XaX  /2 

which,  if  it  is  desired  to  eliminate  s,  may  be  written 

_  8(n2  -  1)W 
"    3  X  a  X  S2  (147J 

It  may  be  observed  that,  for  a  given  material  and  limiting 
tension,  the  required  reduction  in  temperature  is  proportional  to 

fnz  ]\l2 

(n2  —  1)  X  Z2,  or  t  =  ~—j£ '  where  K  is  a  numerical  constant. 

By  using  the  data  for  materials  previously  given  and  assuming 
maximum  allowable  tensions  corresponding  to  S  =  28,000  for 
copper;  13,000  for  aluminum;  and  25,000  for  steel  guy  wire, 
the  calculated  value  of  K  is, 

For  copper  K  =  12,000 
For  aluminum  K  =  38,000 
For  steel  K  =  8,250 

139.  Sag-temperature  Calculations  with  Supports  at  Different 
Elevations. — We  shall  consider  (1)  the  case  of  a  small  difference 
of  elevation  between  supports,  which  will  cause  the  lowest  point 
in  the  span  to  be  below  the  level  of  the  lower  point  of  support, 
as  illustrated  in  Fig.  105;  and  (2)  the  case  of  a  line  running  up  a 
steep  incline,  which  will  cause  the  lower  support  to  be  the  lowest 
point  in  the  span  (see  Fig.  106).  The  problem  will  be  studied  by 
working  out  numerical  examples. 

Data  for  numerical  examples: 

Wire;  No.  0  B.  &  S.  solid  copper. 

Diameter  of  wire,  d  =  0.325  in. 

Cross-sectional  area  of  wire,  A  =  0.083  sq.  in. 

Weight  of  wire  per  foot  run,  w  =  0.32  Ib. 

Distance  between  points  of  support  (measured  on  incline), 
V  =  240  ft. 

Breaking  stress  (say)  55,000  Ib.  per  sq.  in. 

In  these  examples,  we  shall  neglect  ice-coating  on  the  wires, 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  291 


but  allow  for  a  high  wind  velocity  (78  miles  per  hour)  giving 
15  pounds  per  square  foot  of  projected  surface  of  the  wire,  and 
we  shall  also  adopt  the  abnormally  high  factor  of  safety  of  5  as 
called  for  in  the  regulations  issued  by  the  British  Board  of  Trade. 
The  maximum  allowable  stress,  at  a  specified  temperature  t0 
=  22°  F.,  is  therefore 

s       ^55,000 
5 


=  1  ]  ,000  Ib.  per  sq.  in. 


Maximum  tension  in  wire,  Pmax.  =  11,000  X  0.083  =  915  Ib. 

0  325 

The  wind  pressure  per  foot  run  of  the  wire  is  p  =  15  X  — vs- 

12 

=  0.406  pounds,  and  the  value  of  the  factor  n  as  defined  in 
Article  133,  is 


V(0.32)2  +  (0.406) ' 
0.32 


1.58 


FIG.  105. — Wire  hung  between  supports  at  different  elevations. 

For  the  calculation  of  the  critical  temperature  (see  Article  135) 
we  have,  M  =  15  X  106  and  a  =  9.6  X  10~6,  whence,  by  formula 
(143), 

(tc  t0)    =    n  c  vx  i  c  vxirkfi  (   1     ~   T~58/ 


28 


9.6X15X106' 

and  tc  =  28  +  22  =  50°  F.  It  is  at  this  temperature  that  the 
length  of  the  wire  hanging  in  still  air  will  be  the  same  as  the 
length  under  the  condition  of  maximum  load  when  the  stress  is 
n  times  as  great.  It  follows  that  the  tension  in  the  wire  at  a 
temperature  of  50°  F.  (without  wind)  will  be 

Poi  P\ 
max.  »-lO  __Q  ,. 

£  c  =   —         ~    =    < — x~^    =    O/O  ID. 

n          l.oo 

Case  (1).  Small  Difference  of  Elevation.     (See  Fig.  105.)     For 
the  additional  data  required,  assume: 


292  ELECTRIC  POWER  TRANSMISSION 

Difference  in  elevation  of  points  of  support,  h  =  13  ft.,  whence 

1  O 

sin  e  =  240  =  °-0542>  and  cos  e  =  °-9985>  which  indicates  that 
the  span  measured  horizontally,  being  I  =  lr  cos  6,  is  so  nearly 
equal  to  I'  that  no  appreciable  error  will  be  introduced  if  we 
neglect  to  take  account  of  this  difference.  In  this  particular 
example  it  is  therefore  a  matter  of  indifference  whether  we  put 
I  or  I'  in  the  formulas. 

Now  choose  two  or  three  values  of  tension  in  addition  to  the 
tension  Pc  =  578  at  the  "critical"  temperature  of  50°  F.,  and 
calculate  the  corresponding  temperatures  by  the  step  by  step 
method  as  followed  in  the  accompanying  table. 

This  method  of  procedure  is  correct  except  for  the  fact  that 
formula  (123)  used  for  obtaining  item  (/)  does  not  include  all  the 
terms  necessary  for  the  exact  calculation  of  the  length  of  a  para- 
bolic arc.  With  a  difference  in  level  of  only  13  feet  with  supports 
spaced  240  feet  apart,  the  error  is  probably  negligible;  but  with 
a  greater  difference  in  elevation  between  the  points  of  support 
(as  in  Case  (2)  about  to  be  considered)  additional  terms  would 
have  to  be  included  in  the  equation,  thus  adding  to  the  time 
required  for  the  calculations. 

The  maximum  deflection  (item  (c))  of  the  wire  from  the  straight 
line  AB  joining  the  points  of  support  is  always  small  relatively 
to  the  distance  I'  =  AB]  and  in  computing  the  length  of  a  curve 
which  approximates  to  a  straight  line  no  appreciable  difference 
will  be  observable  whether  we  consider  the  curve  to  be  part  of  a 
parabola,  or  catenary,  or  ellipse,  or  circle.  For  our  purpose 
it  will  be  most  convenient  to  calculate  the  length  (X)  of  the 
wire  by  using  formula  (122)  of  Article  127,  merely  substituting 
s'  (item  (c))  for  s,  and  I'  (see  Fig  105)  for  I.  This  has  been 
done  in  the  example,  Case  (1),  the  results  being  given  under  items 
(/)'  and  (gY  at  the  end  of  the  accompanying  table.  The  change  in 
length  (X  —  Xc)  is  seen  to  be  the  same  whichever  method  of  calcu- 
lation is  used,1  and  since  the  latter  method  of  calculating  X 
is  much  shorter  than  the  former,  it  should  be  adopted  when 
making  sag-temperature  calculations  for  spans  on  an  incline. 

Case  (2).  Large  Difference  of  Elevation.  (See  Fig.  106.)  For 
additional  data  required,  assume: 

1  The  very  small  difference  in  the  two  sets  of  figures  may  be  attributed 
to  errors  in  reading  the  slide  rule. 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  293 


OJ                    <M    OS              CO       (N 
CO                    CO                    O                    t^   i-l              O      C<»       >C      "3 

OS  C<l 

o       o         S            S            ~             °°         <f    °    $    SJ 

d  d 

Tj(                Tf 

§2  £ 

cog 

O      O      1C      »O 

<N   O 

>c             d             do         d    o    <N    <N 

d  d 

O           O              "^                    CO                                                                                           CO       OO 

*C                             ^                                            ""!                    *^ 

s 

IIOCCO                   <N                   O                   OOOOOO 
0            t^                                             W                                                                                                                      1C 

d  o 

(X*        ^ 

^                  IN   iO            «*<      OS 
»O                                            <M                    O5   CO              JH       rt< 
1C                                         O                   "^OOOC^OO 

rt*   CO 

||             CO                   0                   0                   00             00      JH      00 

d  d 

1 

•^    :  ii      :            :            ;            ;    ;  -p     :          ;     ; 

I   :  v«     :            :                       :    :  |     :          :     : 

| 

': 

a,    :  ^                :                       :    :    .     :               : 

1 

S$    : 

i«    ij                !                       :    :g     :               : 

• 

_^_   + 

*  jt    *  fj                    •*                 i  1    "    *  J§      •      •            • 

1 

11 

*  p^1    '  ^2                                      **    "^    *     '  *"3          Ctf* 

ii     ii 

i  I  ifi                 V         i  7  f 

1 

'       ®            '                                                                                                               ^-v                     '           '                                      '^x        ^>> 

^ 

^H 

:  .  j    :    •                              «  ^  :    :  g     :  a  -g     : 

i-H 

3   •  ^              -g              +  i    :      §        !    | 

^efl 

•  a    •  &            °J               «,  «    :    :  fe     :    »   .§     : 

•9 

'-   :  .a        £  <N  -             ^  —  i  :    :        :    I    g     : 

:    <*>       •  ^                vx    vx  ft-  „                           \w         •       •     rj          -      -g       8 

(^tc'-fj              S^QS         '^-ii'o     '2°* 

I 

"^     §       '   *bfi              CO   ^  ^                   ~*^         ~|"      ^       '    "M          '       Q-      °5 

.s 

flQn-'S          '"'d'      °°«S      «>     §    '  '**      '    §    w 

«.!•!•             1      1  oS  ^%:  aT    §    o    ''•  *  ^  :§  2    "      ^ 

i 

ji'f  i|  i;7  t  r  |  ?.'  i^i  §  ?  s 

i 

w  ff   *  S                                                 '      ^-^  j3     ii 

Q**    £        .   ^O          •         t>^3                  02                        X                  *     O  '^       "   O                 -|- 

«0 

rf  8,   :  S     :     »        ^           »         ^«C|<^°^:'<S 

1 

1-!  y.|  1    |  ifelvs 

%» 

•I 

"^SiBfl^          S                               |         V~v'ia,£>Vi^    + 

1         : 

I  §  5  •§    1      o         o            o           5  5    9  **  ** 

Iils§^*«        ffi           m      '><SSS^1 

1    " 

CQ             X 

'e  ^    S        S       S         S     3S       3  S  S 

S      3 

294 


ELECTRIC  POWER  TRANSMISSION 


Difference  in  elevation  of  points  of  support,  h  —  140  feet, 

140 
whence  sin  0  =  ^  =  0.583,  and  cos  0  =  0.8124. 

The  span  measured  horizontally  is  I  =  I'  cos  6  =  195  feet. 

If  the  formula  (119)  is  used  for  calculating  the  distance  1A  from 
the  lower  support,  A,  to  the  point  where  the  wire  becomes 
horizontal,  this  quantity  will  be  negative.  It  will  determine  the 
shape  and  position  of  an  imaginary  parabolic  curve  as  indicated 
by  the  dotted  line  of  Fig.  106.  The  vertical  component  of  the 
tension  at  the  point  B  will  be  equal  to  the  weight  of  the  wire  in 


Tension  in  wire 


FIG.  106. — Transmission  line  on  steep  incline. 

the  parabolic  arc  BD;  while  the  vertical  component  at  A  will 
be  the  weight  of  the  portion  AD.  The  horizontal  component 
(Pk)  of  the  tension  will  be  P  cos  6,  where  P  stands  for  the  tension 
at  the  middle  of  the  span  AB. 

Temperature-stress  calculations  can  be  made  by  considering 
the  length  of  wire  in  the  span  AB  to  be  the  difference  between 
the  imaginary  half  spans  BD  and  AD,  the  procedure  being  exactly 
as  indicated  in  the  foregoing  table  illustrating  the  method  for 
small  differences  of  level.  This  method  is  objectionable  because 
the  required  lengths  are  very  small  differences  between  compara- 
tively large  quantities,  and  unless  several  additional  terms  are 
added  to  the  approximate  formula  for  the  length  of  the  parabolic 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS   295 

curve,  serious  errors  may  be  introduced.  We  shall  therefore 
adopt  the  alternative  method,  and  make  the  necessary  calcula- 
tions on  what  may  be  thought  of  as  an  equivalent  horizontal  span. 
Fig.  107  is  an  enlarged  view  of  the  span  AB  of  Fig  106.  The 
loading  of  the  wire  in  a  direction  perpendicular  to  its  length 
is  no  longer  w  Ib  per  foot,  but  w  cos  6  Ib.  per  foot,  and  the  funda- 
mental equation  s  =  ^-  (see  formula  (101)  of  Article  125) 


8Ph 


becomes 


I'Yw  cos  0 
8P 


(148) 


FIG.   107. — Enlarged  sketch  of  span  AB  of  Fig.  106. 


which  can  also  be  written 


SP  cos  e 


this  last  being  the  formula  (122)°  as  developed  in  Article  127. 
The  slope  of  the  wire  being  considerable,  it  will  be  advisable 
to  re-calculate  the  value  of  the  factor  n  which,  in  Case  (1),  was 
taken  to  be  the  same  as  if  the  span  were  horizontal.  The  wind 
pressure  at  right  angles  to  the  wire  is  the  same  as  in  the  pre- 
ceding example,  namly  0.406  Ib.  per  foot;  and  by  definition 
(see  Article  133)  the  value  of  n  is  therefore 


V  (0.32  cos  0)2  +  (0.406)2 


t  fi. 

=  L85 


296  ELECTRIC  POWER  TRANSMISSION 

instead  of  the  previously  calculated  value  of  1.58  which  is  correct 
for  a  horizontal  or  nearly  horizontal  span. 

The  maximum  allowable  tension  being  the  same  as  before,  we 
have 


For  the  "critical"  temperature,  we  have,   by  formula  (143) 

11,000  x  IP*   /        M  _ 

~  9.6  X  15  X  106  V       l.85>  ~ 

whence  tc  =  35  +  22  =  57°  F. 

We  are  now  in  a  position  to  proceed  as  in  Case  (1)  (alternative 
method),  or  if  preferred  we  may  use  the  formulas  of  Article  134 
which  will  give  exactly  the  same  results. 

Let  the  assumed  values  of  tension,  in  addition  to  the  "critical" 
value  Pc  =  494  lb.,  be  PI  =  600  lb.,  and  P2  =  400  Ib.  The 
corresponding  maximum  deflections  from  the  straight  line,  as 
calculated  by  formula  (148)  are, 

s'e  =  3.79  ft. 
s'i  =  3.12ft. 
s't  =  4.68  ft. 

By  formula  (139),  KI  =  8  (3.79)2  =  115 

By  formula  (142),  K2  =  3  X  9.6  X  1Q-6.X(240)2  =  1.66 

494 
By  formula  (141),  K3  =  aQ83  x  9.6  x  15  =  41.5 

By  formula  (138),  putting  s  =  3.12  ft.,  we  have, 

(«7      n       H5-8(3.12)2  /3.79 

(57~'l):         ~L66~  4l'5(^T2 

whence  ti  =  24.7°  F. 

Similarly,  when  s  =  4.68 

*2  =  101°  F. 

The  curve  marked  (2)  in  Fig.  108  has  been  plotted  from  these 
results,  while  curve  (1)  refers  to  the  previously  calculated  Case 
(I).  The  correction  for  n,  and  therefore  for  the  "critical" 
temperature  tc,  need  not  be  made  when  the  slope  is  small.  It  is 
only  when  the  difference  between  the  horizontal  spacing  (I)  and 
the  actual  distance  between  supports  (I')  is  appreciable  that  the 
correction  need  be  made.  The  procedure  here  recommended 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  297 

for  constructing  sag-temperature  curves  for  lines  on  an  incline 
consists  in  replacing  the  actual  span  by  an  "equivalent"  span 
with  supports  on  the  same  level.  This  equivalent  span  may  be 
defined  as  a  horizontal  span  of  the  same  length  (lr)  measured 
between  points  of  support  as  the  actual  span,  the  loading  being 
w  cos  6  Ib.  per  foot  run,  where  w  is  the  weight  per  foot  of  the  wire, 
and  Q  is  the  angle  between  the  direction  of  the  line  and  the 
horizontal. 


130 


Case(2) 
pan  on  considerable 


/400' 

Cased) 

Span  Horizontal 
on  Slight  Incline 


Note    The  figures  on  the  curves 
indicate  the  tension  in  tha 
wire  in  Ibs. 


3.5  4  4.5  5  5.5  6 

Sag,  01  Maximum  Deflection  from  Straight  Line  •  Feet 


6.5 


FIG.  108.  —  Sag-temperature  curves  illustrating  numerical  example. 

The  dotted  curve  marked  (3)  in  Fig.  108  is  plotted  from  the 
same  data  as  curve  (1)  except  that  the  effect  due  to  elasticity  of 
the  wire  has  been  neglected  (item  (h)  in  the  table  on  page  293). 
The  omission  is  often  permissible,  especially  on  long  spans  of 
aluminum  wire;  but,  on  comparatively  short  spans  of  copper 
wire,  as  in  the  present  instance,  it  is  seen  to  give  entirely  inaccu- 
rate results.  The  point  is  mentioned  here  because  of  the  peculi- 
arity that  in  England  —  where  the  stringent  regulations  involving 
large  factors  of  safety  necessitate  uneconomical  construction 
with  short  spans  —  it  is  by  no  means  uncommon  to  neglect  the 
elasticity  of  the  wire;  while  in  America  —  where  the  longer  spans 


298  ELECTRIC  POWER  TRANSMISSION 

often  cause  the  inclusion  of  this  item  to  become  an  unnecessary 
refinement — it  is  customary  to  take  account  of  the  changes  in 
length  due  to  variation  of  load. 

140.  Length  of  Spans.  Conductor  Materials — Copper ;  Alumi- 
num ;  Iron. — When  the  wires  of  a  transmission  line  are  supported 
on  single  wood  poles,  the  span  may  be  anything  from  120  to  250  or 
even  300  ft. ;  but  extra  long  poles,  specially  selected  to  withstand 
the  greater  stresses,  are  required  when  the  span  is  appreciably  in 
excess  of  200  ft. 

On  steel  tower  lines,  very  much  greater  spans  may  be  used. 
The  length  of  span  is  determined  not  only  by  strength  considera- 
tions, but  also  by  considerations  of  economy  (see  Articles  15 
and  16  in  Chapter  III).  The  requirements  in  the  matter  of 
supporting  poles  or  structures  (which  depend  largely  on  length 
of  span)  will  be  referred  to  in  the  following  chapter. 

Whether  copper  or  aluminum  should  be  used  on  a  given  trans- 
mission line  cannot  be  determined  on  general  principles.  Con- 
ductor materials  were  discussed  in  Chapter  IV,  and,  apart  from 
the  physical  properties  of  these  materials,  the  relative  cost, 
which  is  a  variable  quantity,  must  be  taken  account  of  when 
deciding  upon  the  material  best  suited  for  the  work. 

Generally  speaking,  the  deflections  or  sags  of  aluminum  con- 
ductors on  spans  of  moderate  length  will  be  about  30  per  cent:  or 
35  per  cent,  greater  than  with  copper  conductors.  The  difference 
will  be  more  marked  with  the  smaller  sizes  of  wires  and  the  shorter 
spans.  With  extra  long  spans  in  the  neighborhood  of  1000  ft., 
it  will  be  found  that  the  maximum  sag  of  aluminum  and  copper 
conductors  will  be  about  the  same,  if  storm  and  abnormal  winter 
conditions  are  neglected;  that  is  to  say  when  the  factor  n  (defined  in 
Article  134)  is  unity.  Under  this  condition  it  will  even  be  found 
that  copper  has  a  greater  sag  than  aluminum  on  the  very  long 
spans.  The  reason  for  this  is  that  the  greater  temperature  elon- 
gation of  aluminum  is  inappreciable  in  the  case  of  long  spans  with 
necessarily  large  sags,  while  it  is  an  important  factor  in  the  com- 
parison of  the  two  metals  when  the  spans  are  short.  The  above 
statement  is  made  rather  as  being  of  scientific  interest  than  of 
practical  utility,  because,  under  storm  conditions  (when  n  has  a 
large  value),  aluminum  will  be  found  to  be  an  unsatisfactory 
material  to  use  on  long  spans. 

Although  the  tension  in  a  conductor  can  always  be  kept  reason- 
ably small  by  allowing  sufficient  sag,  it  is  obvious  that  a  large  sag 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  299 

involves  higher  and  more  costly  supporting  structures  if  the  clear- 
ance between  ground  and  lowest  point  of  the  conductor  is  to 
remain  the  same.  When  crossing  open  country,  the  clearance 
above  ground  need  not  be  very  great;  but  in  crossing  roads,  a 
clearance  of  21  feet  should  be  allowed,  and  over  foot  paths  the 
clearance  should  never  be  less  than  15  ft.  For  mechanical  rea- 
sons, it  is  a  general  rule  that  no  conductor  for  the  transmission 
of  electric  energy  shall  have  a  lower  ultimate  strength  than  a  No. 
6  B.  &  S.  gauge  hard-drawn  copper  wire. 

Although  it  is  convenient  in  sag  calculations  to  assume  rigid 
supports  at  each  end  of  the  span,  the  deflection,  under  heavy  load, 
of  pole  or  steel  mast  as  used  on  the  shorter  spans,  may  be  re- 
garded to  some  extent  as  a  factor  of  safety.  Refinements  of 
calculation  are  out  of  place  when  figuring  on  short  span  lines :  the 
tendency  is  to  string  wires  of  short-span  transmission  lines  too 
slack  rather  than  too  tight.  On  spans  not  exceeding  150  ft.,  if 
the  wires  are  strung  at  comparatively  low  temperatures,  it  is 
almost  impossible  to  draw  them  up  too  tight. 

The  writer  is  of  the  opinion  of  Mr.  H.  W.  Buck  and  one  or  two 
other  eminent  and  experienced  engineers,  who  hold  that  most 
transmission  lines  are  too  slack.  The  fact  that  the  natural  stretch 
which  takes  place  in  all  overhead  conductors  during  the  few 
months  following  erection,  increases  the  sag  which  is  usually 
excessive  in  the  first  instance,  is  frequently  overlooked;  or,  if 
it  is  recognized,  the  additional  sag  is  erroneously  supposed  to 
increase  the  factor  of  safety.  It  is  not  good  engineering  prac- 
tice to  have  the  wires  hanging  in  festoons  between  the  supports, 
and  the  danger  of  short  circuits  resulting  from  slack  conductors 
being  thrown  together  in  stormy  weather  is  unquestionably  greater 
than  the  risk  of  breakages  on  a  well-constructed  line  with  wires 
strung  tightly  between  towers  of  sufficient  strength  to  withstand 
the  maximum  loads  that  they  may  be  subjected  to  under  the 
worst  weather  conditions. 

EXAMPLES  OF  EXTRA  LONG  SPANS 

Where  power  lines  cross  rivers  or  other  navigable  waterways, 
exceptionally  long  single  spans  are  sometimes  necessary.  The 
longest  in  the  world  crosses  the  St.  Lawrence  river  about  20  miles 
from  Three  Rivers,  Quebec,  and  measures  4800  feet.  The  ends  of 
the  span  are  attached  to  steel  towers  350  feet  high.  Construc- 
tion details  will  be  found  in  the  paper  by  Mr.  S.  Svenningson, 


300  ELECTRIC  POWER  TRANSMISSION 

Proc.  A.  I.  E.  E.,  Nov.,  1918.  Another  long  single  span  is  the 
Missouri  river  crossing  of  the  Mississippi  River  Power  Co.  which 
is  3182  feet  long:  the  conductors  consist  of  copper  wires  of  a 
total  cross-section  of  0.162  sq.  in.  laid  on  a  central  steel  core  of 
0.275  sq.  in.  cross-section. 

The  Pacific  Light  and  Power  Corporation  have  a  2871-foot 
span  at  Sunland,  Cal. :  in  this  case  the  conductors  are  of  aluminum 
wire  of  0.475  sq.  in.  total  cross-section  laid  on  a  central  steel  core 
of  0.062  sq.  in.  cross-section. 

The  Great  Western  Power  Co.  at  Oakland,  Cal.,  have  a  2740-ft. 
span  over  the  navigable  waterway:  it  consists  of  copper-clad 
steel  conductors,  of  0.132  sq.  in.  cross-section.  The  Niagara 
River  crossing  of  the  Canadian  Niagara  Power  Co.  is  2192  ft. 
long:  it  consists  of  copper-clad  steel  conductors  0.155  sq.  in.  in 
cross-section. 

The  longest  existing  span  using  aluminum  cables  without  steel 
core  is  at  Piedra,  Cal.,  on  the  system  of  the  San  Joaquin  Light 
and  Power  Co.:  it  is  1700  feet  long;  the  conductor  cross-section 
being  0.132  sq.  in. 

The  possibilities  of  iron  wire  as  a  conducting  material  for  over- 
head power  lines  was  considered  in  Articles  46  to  49  of  Chapter 
IV.  It  is  only  when  the  price  of  copper  and  aluminum  is  abnor- 
mally high  that  iron  as  a  conductor  material  need  be  seriously 
considered;  but  in  certain  cases — such  as  short  lines  transmitting 
small  amounts  of  energy — the  saving  resulting  from  the  use  of 
iron  wire  may  be  considerable.1  This  is  due  partly  to  the  fact 
that,  with  conductors  of  %  in.  or  %6  in.  galvanized  steel  strand 
cable,  the  spans  may  safely  be  made  longer  than  when  the  equiva- 
lent copper  wire  is  used.  Thus,  for  the  purpose  of  transmitting 
from  50  to  80  kw.  a  distance  of  20  to  30  miles  by  a  single  three- 
phase  line,  the  average  span  on  a  straight  run  across  country 
might  be  300  feet  with  30-ft.  poles,  500  ft.  with  35-ft.  poles,  or 
(preferably)  600  to  700  feet  with  40-  to  45-ft.  poles;  and  the  cost  of 
such  a  line — excluding  the  cost  of  wire — 'might  with  care  and  proper 
supervision  be  no  more  than  two-thirds  of  the  figure  indicated  by 
the  lower  curve  of  Fig.  18  (Chapter  III). 

Wood  poles  will  usually  be  found  satisfactory  for  this  cheap 

1  Useful  mechanical  data  referring  to  iron  and  steel  conductors  will  be 
found  in  an  article  by  Messrs.  Oakes  and  Sahm  in  the  Electrical  World  of 
Aug.  10,  1918,  vol.  Ixxii,  p.  249. 


MECHANICAL  PRINCIPLES  AND  CALCULATIONS  301 


type  of  construction,  but  with  the  longer  spans  it  is  probable  that 
steel  poles  or  masts  handled  and  erected  in  one  piece,  with  or 
without  concrete  settings,  will  frequently  prove  more  economical 
than  wood.  Double-galvanized  wire  or  cable  should  be  used. 
For  very  long  spans  it  may  be  advisable 
to  use  special  high-grade  steel  in  order 
to  avoid  excessive  sag  and  spacing  be- 
tween wires,  and  to  pay  particular  at- 
tention to  the  guying  of  corner  poles. 

141.  Factors  of  Safety.  Joints  and 
Ties.  —  The  factors  of  safety  usually 
adopted  when  calculating  the  permis- 
sible tension  in  overhead  wires  have 
already  been  referred  to  in  Article  131, 
and  little  remains  to  be  added  here. 
In  America,  the  factor  of  safety  for 
conductors  is  about  2  :  this  means  that, 
under  the  worst  assumed  conditions  of 
loading,  the  material  of  the  conductor 
may  be  stressed  to  very  nearly  its 
elastic  limit.  It  is  usual  to  assume  a 
wind  velocity  of  70  miles  per  hour  com- 
bined with  a  sleet  deposit  H  in.  thick 
at  a  temperature  of  0°  F.  ,  For  guy 
wires,  a  factor  of  safety  of  3  to  3>^  is 
generally  allowed. 

In  Great  Britain,  where  higher  fac- 
tors of  safety  are  used,  the  Board  of 
Trade  calls  for  a  maximum  tension  not 
exceeding  one-fifth  of  the  breaking  load, 
on  the  assumption  of  a  temperature  of 
22°  F.  and  a  wind  velocity  correspond- 
ing to  a  pressure  of  15  Ib.  per  square 
foot  of  the  projected  surface  of  the 
wire.  Possible  accumulations  of  snow 
and  ice  are  ignored. 

On  the  Continent  of  Europe,  the  net  factor  of  safety  for  wires 
under  the  worst  conditions  of  loading  is  about  2^. 

When  the  "flexible"  type  of  tower  construction  is  used,  it  is 
customary  to  allow  a  somewhat  higher  factor  of  safety  for  the 
conductors  than  when  the  towers  are  of  the  so-called  rigid  type. 

UEJNAHY 


FIG.    109. — Type    of    joint 
for  overhead  conductors. 


(0*5' 


302  ELECTRIC  POWER  TRANSMISSION 

Joints  in  wires  can  and  should  be  made  of  the  same  strength  as 
the  wire  itself.  A  discussion  of  the  various  types  of  joints  as 
used  in  practice  would  be  out  of  place  in  these  pages.  A  very 
common  type  is  the  Mclntyre  joint,  illustrated  in  Fig.  109. 
It  is  claimed  that  when  the  sleeve  is  long  enough  to  allow  of 
three  complete  turns,  the  strength  of  this  joint  is  equal  to  that  of 
the  cable  itself.  Under  favorable  weather  conditions  it  is  pos- 
sible to  make  very  good  butt- welded  joints  on  aluminum  con- 
ductors with  the  aid  of  a  blow  lamp.  Such  joints  would  probably 
not  be  satisfactory  on  conductors  of  large  cross-section  (more  than 
%  in.  diameter),  and  they  should  be  made  at  the  insulator  where 
they  will  not  be  subjected  to  the  maximum  tensile  stress. 

The  tie  wire  usually  secures  the  conductor  to  the  insulator  in 
such  a  way  as  to  prevent  as  far  as  possible  the  creeping  of  the  con- 
ductor from  one  span  to  another.  In  some  cases,  however,  it  is 
advisable  to  allow  the  conductor  to  slip,  or  the  tie  wire  to  break, 
before  the  tension  in  the  cable  is  great  enough  to  damage  insulator 
or  cross-arm.  Soft  or  semi-hard  wire  is  generally  used  for  ties,  and 
the  size  should  not  be  less  than  No.  4  B.  &  S.  copper,  or  No.  2 
B.  &  S.  aluminum.  The  tie  wire  should  be  of  the  same  material 
as  the  conductor. 

Some  forms  of  tie  include  a  serving  of  the  tie  wire  extending 
an  appreciable  distance  on  each  side  of  the  insulator;  the  object 
being  to  afford  some  protection  to  the  conductor  in  the  event 
of  an  arc  striking  over  the  insulator. 

A  long  sleeve  over  line  wires  at  point  of  support  is  objection- 
able. It  is  true  that  breakages  of  wires  caused  by  too  great 
rigidity  at  points  of  support  are  of  rare  occurrence;  but  where 
mechanical  forms  of  cable  clamps  are  used,  as  with  the  suspen- 
sion type  of  insulator,  and  in  many  of  the  larger  pin  type  insu- 
lators, the  method  of  clamping  the  conductor  to  avoid  mechanical 
injury  or  weakening  due  to  vibration  or  to  swaying  of  the  wires  in 
a  wind,  should  receive  careful  attention. 


CHAPTER  X 
TRANSMISSION-LINE  SUPPORTS 

142.  General  Considerations.  Types  of  Transmission-line 
Supports. — The  supporting  poles  or  structures  for  overhead 
electric  power  transmission  lines  are  of  various  kinds.  Where 
the  ordinary  wood  telegraph  pole  or  a  larger  single  pole  of  similar 
type  is  not  suitable,  double  poles  of  the  "A"  or  "H"  type,  or 
even  braced  wooden  towers  of  considerable  height  and  strength, 
may  sometimes  be  used  with  advantage.  Under  certain  condi- 
tions it  may  be  economical  to  use  steel  poles  of  the  tubular  type, 
or  light  masts  of  latticed  steel,  even  for  comparatively  short  spans; 
and  poles  of  reinforced  concrete  have  much  to  recommend  them. 
But  for  long  spans,  and  the  wide  spacing  of  wires  necessary  with 
the  higher  pressures,  steel  towers,  either  of  the  rigid  or  flexible  type, 
will  generally  be  required. 

The  use  of  wood  poles  is  therefore  limited  to  comparatively  low 
pressures,  and  spans  of  moderate  length.  It  is  probable  that,  in 
rough  country  where  suitable  timber  is  plentiful,  and  the  cost  of 
transporting  steel  towers  would  be  high,  wooden  supports  of  the 
"A"  or  "H"  type  might  be  used  economically  for  voltages  up  to 
60,000,  even  if  two  three-phase  circuits  were  carried  on  one  set  of 
poles;  but  for  higher  voltages  the  steel  tower  construction  with 
fairly  long  spans,  would  in  almost  every  case,  be  preferable. 
The  case  of  a  100,000-volt  three-phase  line  supported  on  wood 
poles  was  mentioned  in  Chapter  I  as  an  example  of  economic 
wood-pole  construction  for  a  high-voltage  transmission.  Another 
example  of  a  wood  structure,  erected  where  steel  would  have  been 
adopted  by  many  engineers  as  being  the  only  material  available, 
occurs  at  a  river  crossing  between  Astoria  and  Flavel,  Ore. 
At  this  point  a  1611-foot  span  is  supported  on  wood-pole  towers 
135  ft.  high,  a  description  of  which  will  be  found  in  the  Electrical 
World  oi  Sept.  11,  1915. 

The  decision  as  to  the  best  type  of  line  to  adopt  is  not  easily 
or  quickly  arrived  at.  The  problem  is  mainly  an  economic  one, 
and  the  decision  will  depend,  not  only  on  the  first  cost  of  the 

303 


304  ELECTRIC  POWER  TRANSMISSION 

various  types  of  line  construction,  but  also  on  the  probable  life 
of  the  line  and  the  cost  of  maintenance. 

It  is  necessary  to  make  up  many  preliminary  estimates  of  the 
completed  line,  and  these  must  obviously  include,  not  only  the 
cost  of  the  various  types  of  supporting  structure  delivered  at 
points  along  the  line,  but  also  the  cost  of  foundations  and  erection. 
Again,  even  if  a  suitable  kind  of  wood  is  readily  available  in  the 
district  to  be  traversed  by  the  transmission  line,  it  is  possible  that 
the  cost  of  seasoning  the  poles,  and  treating  them  with  preserva- 
tive compounds  to  ensure  a  reasonably  long  life,  may  render  the 
use  of  steel  structures  more  economical  even  for  comparatively 
low  pressures.  The  use  of  latticed  steel  poles,  from  30  to  40  feet 
high,  capable  of  being  shipped  and  handled  in  one  piece,  appears 
to  be  gaining  favor  in  districts  where  ultimate  economy  over  wood 
poles  can  be  shown  to  result  from  the  adopcion  of  these  light  steel 
structures.  The  life  of  a  steel  tower  line  depends  somewhat  on 
climatic  conditions.  In  Great  Britain  the  dampness  of  the 
climate,  together  with  the  impurities  in  the  atmosphere  in  the 
neighborhood  of  manufacturing  and  populous  districts,  render 
light  steel  structures  less  durable  than  in  America  (except, 
perhaps,  on  the  Pacific  coast,  where  special  precautions  are  re- 
quired to  guard  against  rapid  corrosion  due  to  the  prevalence  of 
fogs  and  moisture).  Not  only  has  the  iron- work  protected  by 
paint  to  be  repainted  on  the  average  every  three  years,  but  the 
spans  must  usually  be  short,  as  the  private  ownership  of  valuable 
property  renders  the  construction  of  a  straight  transmission 
line  with  long  equal  spans  almost  impossible  in  the  United 
Kingdom.  These  conditions  are  all  in  favor  of  the  employment 
of  selected  and  well  creosoted  wood  poles,  the  life  of  which  may 
be  30  years  or  more. 

When  making  comparisons  between  wood  and  steel  for  trans- 
mission line  supports,  it  is  not  only  the  matter  of  first  cost  that 
has  to  be  considered.  Steel  structures  have  the  advantage  of 
being  invulnerable  to  prairie  and  forest  fires;  moreover,  owing  to 
the  longer  spans  rendered  possible  by  the  stronger  and  taller 
supports,  there  is  less  chance  of  stoppages  owing  to  broken  insula- 
tors, and  less  leakage  loss  over  the  surface  of  insulators.  A 
fact  that  is  often  overlooked  is  that  the  size  of  conductor  limits  the 
practical  length  of  span;  for  instance,  with  a  small  conductor 
such  as  a  No.  4  B.  &  S.,  it  would  not  be  wise  to  have  spans  much 
above  250  or  300  ft.  This  suggests  what  is  frequently  found  to  be 


TRANSMISSION-LINE  SUPPORTS  305 

the  case,  namely,  that  the  total  cost  of  a  line  may  be  reduced  by 
using  a  conductor  of  rather  larger  section  than  the  electrical  cal- 
culations would  indicate  as  being  necessary,  because  the  stronger 
cable  permits  of  a  wider  spacing  of  the  supporting  towers. 

The  economic  length  of  span  on  steel  tower  lines  usually  lies 
between  500  and  700  ft.,  but  very  much  longer  spans  can  be  used 
where  the  character  of  the  country  would  render  then-  use  eco- 
nomical or  where  rivers  have  to  be  crossed.1  On  the  transmission 
system  supplying  Dunedin  City,  New  Zealand,  with  electric 
energy  at  35,000  volts,  there  is  a  span  1700  ft.  long  where  the  line 
crosses  the  ravine  near  the  power  station.  The  peculiarity  of 
this  span  is  the  great  difference  in  level  between  the  two  supports, 
the  upper  tower,  which  is  a  special  steel  structure,  being  650  ft. 
above  the  lower  tower. 

143.  Wood  Pole  Lines.  Kinds  of  Wood  Available. — Among 
the  varieties  of  straight-growing  timber  used  for  pole  lines  on  the 
American  continent  may  be  mentioned  cedar,  chestnut,  oak, 
cypress,  juniper,  pine,  tamarack,  fir,  redwood,  spruce,  and  locust. 
In  England  the  wood  poles  are  usually  of  Baltic  pine  or  red  fir 
from  Sweden,  Norway,  and  Russia.  The  woods  used  for  the 
cross-arms  carrying  the  insulators  include  Norway  pine,  yellow 
pine,  cypress  or  Douglas  fir,  oak,  chestnut,  and  locust. 

Probably  the  best  wood  for  poles  is  cedar;  but  chestnut  also 
makes  excellent  durable  poles.  Much  depends,  however,  on  the 
nature  of  the  soil,  and,  generally  speaking,  poles  cut  from  native 
timber  will  be  more  durable  than  poles  of  otherwise  equally  good 
quality  grown  under  different  conditions  of  soil  and  climate. 

With  the  more  extended  adoption  of  preservative  treatments 
(to  be  referred  to  later),  the  inferior  kinds  of  timber  which 
under  ordinary  conditions  would  decay  rapidly,  will  become  of 
relatively  greater  value,  and  with  the  growing  scarcity  of  the 
better  kinds  of  timber,  it  is  probable  that  poles  of  yellow  pine, 
tamarack,  and  Douglas  fir  will  be  used  more  extensively  in  the 
future. 

The  trees  should  be  felled  during  the  winter  months,  and  after 
being  peeled  and  trimmed  should  be  allowed  to  season  for  a  period 
of  at  least  twelve  months. 

A  brief  specification  covering  wood  poles  for  power  transmis- 
sion lines  will  be  found  in  Appendix  II  at  the  end  of  this  book. 

1  See  Article  140,  Chapter  IX. 
20 


306 


ELECTRIC  POWER  TRANSMISSION 


144.  Typical  Wood  Pole  Lines. — For  a  single  three-phase  line 
transmitting  power  at  20,000  to  22,000  volts  single  poles  having 
a  top  measurement  of  about  8  in.  would  be  suitable.  The  dis- 
tance between  wires  would  be  about  3  ft.;  the  arrangement 
being  as  shown  on  the  sketch,  Fig.  1 10,  with  pole-top  details  as  in 
Fig.  111.  This  shows  an  arrangement  without  overhead  guard 
wire,  but  with  some  or  all  poles  protected  by  a  grounded  light- 


Fio.  110. — Typical  wood  pole  line  for  pressures  up  to  22,000  volts  three-phase. 

ning  rod.  In  exposed  positions,  and  at  angles,  pieces  of  bent 
flat  iron  may  be  fitted  with  advantage  on  the  cross-arm  near  the 
insulators,  as  shown  by  the  dotted  lines  in  Fig.  111.  These 
pieces  serve  the  double  purpose  of  hook  guard  in  case  of  the  wire 
shipping  off  insulator,  and  of  additional  protection  against  light- 
ning. A  discharge  from  the  line  tends  to  leap  across  to  this 
grounded  metal  horn  over  the  surface  of  the  insulator,  thus  fre- 
quently preventing  the  piercing  or  shattering  of  insulators. 
Fig.  112  shows  the  pole-top  arrangement  for  a  33,000- volt 


TRANSMISSION-LINE  SUPPORTS 


307 


line  of  the  Central  Illinois  Company,  while  the  double  insulator 
construction  adopted  by  the  same  company  for  corner  poles  is 
illustrated  in  Fig.  113.  These  two  illustrations  are  reproduced 
here  by  kind  permission  of  the  Electrical  World. 

A  simple  "A"  frame  construction  for  a  duplicate  three-phase 
line  operating  at  11,000  volts  is  shown  in  Fig.  114.  Another 
type  of  construction  for  duplicate  three-phase  line  is  shown  in 
Fig.  115,  where  the  standard  single  wood  pole  is  used.  This 
is  the  arrangement  adopted  by  the  Central  Illinois  Company 
when  it  is  desired  to  carry  two  circuits  on  the  one  pole  line. 


Galvanized  Iron 
Lightning  Rod 


Special  Pin  for 
Pole-Top  Insulator 


%>  Galvanized  Steel  Cable 
connecting  Lightning  Rod 
to  Ground 


FIG.  111. — Pole-top  details. 


145.  Life   of   Wood  Poles     Preservative  Treatment— It  is 

not  easy  to  estimate  the  probable  life  of  poles,  because  this  will 
depend  not  only  on  the  kind  of  timber,  but  also  on  the  nature 
of  the  soil,  climatic  conditions,  the  time  of  seasoning,  whether  or 
not  the  poles  have  received  treatment  with  preservative  com- 
pounds, and  the  nature  of  such  treatment. 

In  England  the  life  of  well-seasoned,  creosoted  poles  may  be 
about  35  years  in  good  soil,  and  from  18  to  20  years  in  poor 


308  ELECTRIC  POWER  TRANSMISSION 


FIG.  112. — Single-circuit  construction  for  three-phase  overhead  transmission. 


FIG.  113. — Double-arm  corner  construction. 


TRANSMISSION-LINE  SUPPORTS 


309 


soil.  On  the  American  continent,  where  untreated  poles  have 
been  used  in  large  numbers,  the  average  life  is  probably  about  12 
years.  The  better  woods,  such  as  cedar  and  chestnut,  might 
last  on  the  average  14  to  16  years,  while  juniper  and  pine  might 
have  to  be  replaced  in  6  to  10  years.  On  certain  lines  where  un- 
treated poles  of  unsuitable  timber  have  been  erected  in  poor  soil, 
or  where  destructive  insects  are  particularly  active,  the  poles 
have  had  to  be  replaced  in  less  than  4  years.  The  creosoted  poles, 
as  used  in  England,  will  usually  stand  best  in  moist  or  clayey 


imber  Tie  About  6x9x5 


LI 


FIG.  114. — Typical  "A"  frame  construction  for  duplicate  three-phase  line. 

ground;  there  is  a  tendency  for  the  creosote  to  run  out  and  be 
absorbed  into  the  ground  when  the  soil  is  loose  and  sandy,  with 
the  result  that  the  poles  deteriorate  rapidly  just  below  the  ground 
level.  Marshy  soil  is  generally  bad  for  wood  poles,  also  ground 
that  is  alternately  wet  and  dry. 

Preservative  Treatment  of  Poles. — Many  chemical  solutions 
and  methods  of  forcing  them  into  the  wood  have  been  tried 
and  used  with  varying  success;  but  it  is  generally  conceded  that 


310 


ELECTRIC  POWER  TRANSMISSION 


treatment  with  coal-tar  creosote  oil  gives  the  best  protection 
against  decay;  and  its  cost  is  probably  lower  than  that  of  any 
other  satisfactory  treatment. 

There  are  three  recognized  methods  of  applying  oil : 

(a)  The  high-pressure  treatment  (Bethel  system). 

(b)  The  open-tank  treatment. 

(c)  Brush  treatments. 

In  Europe  the  treatment  known  as  the  Riiping  process  is 
largely  used;  it  is  less  costly  than  the  Bethel  system.     In  France 


This  Bayonet  for 
CurYft  in  line  '•- 


*4<---jV— -*-- 

Uf 


Fio.  115.  —  Double-circuit  construction. 


copper  sulphate  is  used  extensively  as  a  preservative  (Boucherie 
process),  but  the  results  are  not  entirely  satisfactory. 


HIGH-PRESSURE  TREATMENT 


This  is  undoubtedly  the  best,  but  it  is  also  the  most  costly. 
The  poles,  after  being  trimmed  and  framed,  are  placed  in  large 


TRANSMISSION-LINE  SUPPORTS  311 

treating  cylinders  capable  of  being  hermetically  closed.  If  the 
poles  are  green  or  wet  they  are  first  subjected,  in  these  cylinders, 
to  a  steaming  process  from  three  to  eight  hours,  the  steam  being 
admitted  under  a  pressure  of  12  to  20  Ib.  The  steam  is  then 
blown  off,  and  the  treating  cylinder  is  exhausted,  the  vacuum 
being  maintained  for  a  period  of  one  to  two  hours.  Immediately 
afterward  the  creosote  is  forced  in  under  pressure  at  a  tempera- 
ture of  140°  to  200°  F.  Seasoned  timber  is  not  subjected  to  the 
steaming  process,  but  the  temperature  inside  the  treating  cylinder 
is  raised  by  means  of  heating  coils  to  about  150°  F.  prior  to  the 
filling  process. 

The  poles  will  absorb  from  a  minimum  of  10  Ib.  to  a  maximum 
of  15  Ib.  of  oil  per  cubic  foot.  The  softer  and  more  porous  woods 
will  absorb  the  most  oil;  but  on  the  other  hand,  the  benefit  such 
woods  derive  from  the  treatment  in  the  matter  of  increased  life  is 
more  marked  than  in  the  case  of  the  closer  grained  timber. 


OPEN-TANK  TREATMENT 

The  butts  of  the  poles  are  placed  in  the  creosote  oil,  which  is 
preferably  heated  to  a  temperature  of  200°  F.  to  220°  F.  They 
are  maintained  in  this  bath  for  a  period  of  one  to  three  hours, 
after  which  they  are  placed  in  cold  oil  for  another  period  of  one 
to  three  hours.  This  process  will  permit  of  a  complete  penetra- 
tion of  the  sapwood  to  a  height  of  about  2  ft.  above  ground  level. 
When  properly  carried  out  it  is  capable  of  giving  very  satisfac- 
tory results.  The  open-tank  process  is  specially  applicable  to 
the  treatment  of  the  more  durable  kinds  of  timber,  such  as  cedar 
and  chestnut. 

BRUSH  TREATMENT 

The  oil  is  applied  hot  with  hard  brushes,  a  second  coat  being 
applied  after  the  first  has  soaked  in.  The  temperature  of  the 
oil  should  be  about  200°  F.  This  method  of  application  of  the 
oil  is  the  cheapest  and  the  least  effective,  but  it  affords  some  pro- 
tection when  the  wood  is  well  seasoned  and  dry.  There  is  little 
advantage  to  be  gained  by  the  external  application  of  preserva- 
tive compounds  to  green  timber;  indeed,  the  sealing  up  of  the 
surface  of  such  timber,  by  enclosing  the  fermentative  juices,  may 
even  lead  to  more  rapid  decay.  The  brush  treatment  cannot  be 
applied  to  poles  which  are  set  in  the  winter  months  in  cold  cli- 


312  ELECTRIC  POWER  TRANSMISSION 

mates,  as  the  frost  would  so  harden  the  surface  of  the  poles  that 
there  would  be  no  absorption  of  the  preservative  liquid. 

The  quality  of  oil  used,  whatever  the  method  of  application, 
is  a  matter  of  importance.  In  circular  No.  98  of  the  United 
States  Forest  Service,  Department  of  Agriculture,  issued  May  9, 
1907,  the  concluding  statement  is  to  the  effect  that  light  oils  boil- 
ing below  400°  F.  will  not  remain  in  the  timber;  but  the  heavy 
oils,  containing  a  high  percentage  of  anthracene  oil,  will  remain 
almost  indefinitely,  and  afford  excellent  protection  against  decay 
and  boring  animals. 

The  reader  who  desires  to  go  further  into  the  important  ques- 
tion of  wood  pole  preservation,  is  referred  to  the  excellent  book 
by  Howard  F.  Weiss,  "  The  Preservation  of  Structural  Timber" 
(McGraw-Hill  Book  Company).1  It  seems  hardly  necessary  to 
point  out  that  the  saving  effected  by  prolonging  the  life  of  the 
poles  will  usually  justify  the  cost  of  the  special  treatment. 
Given  reliable  data  regarding  costs  and  probable  life  of  treated 
and  untreated  poles,  the  necessary  calculations  are  easily  made. 
As  an  example;  if  the  cost  of  a  pole  after  treating  by  the  brush 
process  is  $24  and  the  average  life  10  years,  while  the  extra 
cost  of  open-tank  treatment  is  $1  with  an  added  10  years  to  the 
life  of  the  pole,  it  is  easy  to  show  that  the  tank  treatment  will  be 
the  more  economical  in  the  long  run. 

INFLAMMABILITY  OF  TREATED  TIMBER 

Poles  and  cross-arms  treated  with  creosote  oil  are  less  liable  to 
destruction  by  fire  than  untreated  timber  of  the  same  kind. 
This  appears  to  be  due  to  the  fact  that  the  free  carbon  deposited 
by  the  burning  oil  on  the  surface  of  the  timber  affords  some  pro- 
tection from  the  action  of  the  fire.  A  committee  appointed  by  the 
National  Electric  Light  Association  of  New  York  City  conducted 
a  series  of  experiments  on  similar  specimens  of  treated  and  un- 
treated short-leaf  pine,  and  proved  conclusively  that  the  latter 
suffered  considerably  more  damage  from  the  effects  of  fire  than 
the  specimens  that  had  been  impregnated  with  creosote  oil. 

Reinforcing  Pole  Butts. — Wood  poles  usually  decay  most 
rapidly  at  or  near  the  ground  surface.  It  is  not  always  necessary 

1  See  also  "Preservative  Treatment  of  Telephone  Poles,"  by  F.  L.  Rhodes 
and  R.  F.  Hosford,  Trans.  A.  1.  E.  E.,  vol.  xxxiv  (part  2),  p.  2549,  Oct., 
1915. 


TRANSMISSION-LINE  SUPPORTS  313 

to  replace  poles  which  may  be  otherwise  sound,  but  which  have 
been  weakened  locally  by  decay  just  below  the  ground  surface. 
They  may  be  reinforced  by  means  of  steel  rods  about  >£  in.  diame- 
ter, pointed  at  both  ends  and  driven  into  the  pole  above  and 
below  the  damaged  portion.  Concrete  is  then  filled  in  around 
the  pole,  extending  at  least  12  in.  above  the  ground  level.  This 
and  similar  methods  of  prolonging  the  life  of  poles  have  proved 
satisfactory.  The  cost  may  be  from  $3  to  $4  per  pole,  and  the 
life  may  be  prolonged  5  to  10  years. 

146.  Insulating   Qualities  of  Wood  Poles. — One  advantage 
which  may  be  claimed  for  wood  poles  is  the  possibility  of  work- 
ing on  live  wires  with  little  danger  to  life  when  the  conditions  are 
favorable.     In  the  Black  Hills  in  the  mining  district  of  South 
Dakota,  where  the  climate  is  dry,  it  is  usual  to  work  on  live  wires 
at  24,000  volts,  supported  on  ungrounded  poles,  without  using  any 
insulating  devices.     In  this  instance,  the  separation  between 
wires  is  unusually  large,  being  5  ft.,  which  affords  additional 
safety.     On  the  other  hand,  it  may  be  argued  that  accidental 
contact  with  a  wood  pole  carrying  high-tension  conductors  may 
be  a  danger  to  the  public,  which  is  practically  absent  in  the  case 
of  grounded  steel  poles  or  towers.     Tests  have  been  made  to 
determine,  if  possible,  the  nature  of  charges  likely  to  pass  to 
ground  through  a  person  touching  ungrounded  wooden  poles  of 
a  high-tension  transmission.     These  tests  show  that  it  is  pos- 
sible for  poles  to  become  dangerously  charged,  but  not  probable. 
A  grounded  metal  ring  or  wire  placed  around  the  pole  from  6  to  7 
ft.   above  ground  eliminates  all  possibility  of  accidents  from 
this  cause. 

147.  Weight  of  Wood  Poles.— For  the  purpose  of  estimating 
the  costs  of  transport  and  handling  of  poles,  the  weight  may  be 
calculated  on  the  assumption  that  the  pole  is  of  circular  section 
and  of  uniform  taper,  such  that  the  diameter  D  at  the  bottom  is 
equal  to  the  diameter  d  at  the  top  plus  a  quantity  tH,  of  which 
H  is  the  distance  between  the  two  sections  considered,  and  t  is 
a  constant  depending  on  the  taper  and  therefore  on  the  kind  of 
wood.     Some  approximate  (average)  values  of  t  together  with 
average  weight  per  cubic  foot  of  various  kinds  of  dry  timber  will 
be  found  in  the  accompanying  table,  from  which  the  value  of  t 
for  cedar  is  seen  to  be  0.0165  (the  height  H  being  understood  to 
be  expressed  in  inches'),  and  the  weight  per  cubic  foot,  35  lb.; 


314  ELECTRIC  POWER  TRANSMISSION 

CONSTANTS  FOR  WOOD  POLES 


Kind  of  wood 

Wt.  per  cubic 
foot,  Ib. 
approx. 

Natural 
taper  t, 
average 

Modulus  of 
rupture 

Modulus  of 
elasticity 

M" 

Juniper  
American  Eastern 
white  cedar  
Spruce  
White  pine  

35 
27 
26 

0.0165 

3700 

4000 
4500 
4500 

700,000 
1,300,000 
1,000,000 

Red  pine  
Douglas  fir3 

34 
34 

5000 
6500 

1,400  000 

Norway  pine 

7000 

1,400000 

Redwood  
Idaho  cedar  
Chestnut  

23 

42 

0.01 

0.016 

7000 
6000 
6000 

700,000 
900,000 

1  Being  stress  in  pounds  per  square  inch  at  moment  of  rupture  under 
bending  conditions. 

2  Inch  units.     Average  figures,  which  must  be  considered  approximate 
only. 

3  This  name  is  intended  to  cover  yellow  fir,  red  fir,  Western  fir,  Washing- 
ton fir,  Oregon  fir,  North-west  and  West-coast  fir. 

The  volume  of  a  frustrum  of  right  circular  cone  is: 


but 


Volume  = 

o 
D     =  d 


X 

4 


+  Dd  +  d2) 

tH,  and  the  formula  becomes: 
Volume  =         (3d2  +  t*H*  +  UtH) 


By  using  this  formula  and  putting  for  H  the  value  65  X  12  = 
780  in.,  and  for  d  the  value  7  in.,  the  weight  of  a  pole  of  American 
eastern  white  cedar  measuring  7  in.  diameter  at  top  and  65  ft. 
over-all  length  works  out  at  2410  Ib. 

148.  Strength  and  Elasticity  of  Wood  Poles.— Apart  from 
the  dead  weight  to  be  supported  by  the  poles  of  a  transmission 
line — which  will  include  not  only  the  fixtures  and  the  conductors 
themselves,  but  also  the  added  weight  of  sleet  or  ice  in  climates 
where  ice  formation  is  possible — the  stresses  to  be  withstood 
include  the  resultant  pull  of  the  wires  in  adjoining  spans,  and  the 
wind  pressure  on  poles  and  wires.  It  is  customary  to  disregard 
the  dead  weight  or  column  loading,  except  when  the  spans  are 
large  and  the  conductors  numerous  and  heavy.  A  formula  for 
approximate  calculation  .of  loads  carried  by  poles  when  acting  as 


TRANSMISSION-LINE  SUPPORTS  315 

struts  or  columns  will  be  given  later.  The  pull  due  to  the  con- 
ductors on  corner  poles  is  usually  met  by  guying  these  poles,  by 
which  means  the  pull  tending  to  bend  the  pole  is  largely  converted 
into  an  increased  vertical  downward  pressure;  but  even  on  straight 
runs  there  may  be  stresses  due  to  unequal  lengths  of  span  which 
would  cause  a  difference  in  the  tensions  on  each  side  of  the  pole. 
The  most  important  stresses  to  which  the  poles  are  subjected, 
apart  from  such  accidents  as  are  due  to  falling  trees  or  the  sever- 
ing of  all  wires  in  one  span,  are  those  caused  by  strong  winds 
blowing  across  the  line.  The  resulting  pressure  at  pole-top  due 
to  strong  winds  acting  on  long  spans  of  ice-coated  wires  may  be 
very  great,  and  the  poles  must  be  strong  enough  to  resist  this.1 

For  the  purpose  of  making  strength  and  deflection  calculations, 
the  pole  may  be  considered  as  a  truncated  cone  of  circular  sec- 
tion, firmly  fixed  in  the  ground  at  the  thick  end,  with  a  load 
near  the  small  end  in  the  form  of  a  single  concentrated  resultant 
horizontal  pull.  The  calculation  is  therefore  exactly  the  same  as 
for  a  beam  fixed  at  one  end  and  loaded  at  the  other.  Such  a 
beam,  if  it  exceeds  a  certain  length  depending  upon  the  amount  of 
taper,  will  not  break  at  the  point  where  the  bending  moment  is 
greatest  (i.e.,  at  the  ground  level),  because,  in  a  beam  of  circular 
section  and  uniform  taper,  the  stresses  in  the  material  are  not 
necessarily  greatest  at  this  point,  as  will  be  shown  later.  The 
ordinary  telegraph  or  electric  lighting  pole  usually  breaks  at 
a  point  about  5  ft.  above  ground  unless  the  butt  has  been  weak- 
ened by  decay. 

Calculations  on  strengths  and  deflections  of  wood  poles 
cannot  be  made  with  the  same  accuracy  as  in  the  case  of  steel 
structures;  and  the  constants  in  the  table  of  Article  147  are  aver- 
ages only  for  approximate  calculations.  The  factor  of  safety 
generally  used  on  the  American  continent  is  6,  both  for  poles 
and  cross-arms.  The  maximum  wind  pressure  is  taken  at  30  Ib. 
per  square  foot  of  flat  surface,  or  18  Ib.  per  square  foot  of 
projected  surface  of  smooth  cylinders  of  not  very  large  diameter. 
In  England  the  factor  of  safety  for  telegraph  poles  is  8,  and  for 
power  lines  10.  The  latter  figure  would  seem  to  be  unnecessarily 
high:  it  suggests  a  want  of  confidence  either  in  the  strength 
calculations  or  in  the  tests  and  load  assumptions  on  which  the 
calculations  are  based. 

1  See  Article  131  in  Chapter  IX. 


316 


ELECTRIC  POWER  TRANSMISSION 


149.  Calculation  of  Pole  Strengths. — The  relation  between  the 
externally  applied  load  and  the  stresses  in  the  fibers  of  the  wood 
is: 

Bending  moment  =  stress  in  fibers  most  remote  from  neutral 
axis  X  section  factor,  or  MB  =  S  X  Z. 

If  P  is  the  force  in  pounds  applied  at  a  point  distant  x  in.  from 
the  cross-section  A  (see  Fig.  116),  then: 

MB  =  Px  Ib.-in. 

And  if  the  stress  S  is  expressed  in  pounds 
p  per  square  inch,  and  the  section  is  as- 

sumed circular: 

Px  -  S  X  ^ 

JTX    —    >J    A.    ~7T^~ 


"~Dia.  -  d  +  tx    But  it  is  assumed  that  the  diameter  at 
^  any  point  x  in.   below  the  section  of 

diameter  d  is  d  X  tx,  therefore: 


32P 


X 


(d  + 


(149) 


FIG.    116.— Wood   pole    with 
horizontal  load  near  top. 


In  order  to  find  the  position  of  the 
cross-section  at  which  the  pole  is  most 
likely  to  break — that  is  to  say,  where 
the  fiber  stress  is  a  maximum — it  is 
necessary  to  differentiate  the  last  equa- 
tion with  respect  to  x,  and  find  the 
value  of  x  which  makes  this  differential 
equal  to  zero.  This  gives 

d 


for  the  point  where  the  stress  S  is  a  maximum.  The  position  of 
this  cross-section  is  evidently  not  always  at  ground  level.  If 
this  value  of  x  is  greater  than  H,  then  the  maximum  fiber  stress 
will  be  at  ground  level,  and  it  is  calculated  by  substituting  H  for 
x  in  formula  (149). 

The  diameter  of  the  pole  at  the  weakest  point  is: 

&  -  d  +  tx 


=  1.5  d 


TRANSMISSION-LINE  SUPPORTS  317 

and  it  is  only  when  the  diameter  at  ground  level  is  greater  than 
one  and  a  half  times  the  diameter  where  the  pull  is  applied  that 
the  pole  may  be  expected  to  break  above  ground  level. 

If  the  stress  S,  the  taper  t,  and  the  pole-top  diameter  d  are 
known,  the  load  P  is  readily  calculated  as  follows: 

Bending  moment  =  P  X  x 

ird3w 
Resisting  moment  =  S  X  —^ 

Sm 

But  x  =     -  and  dw  =  l.5d 


therefore 


Pd  TT  X 

TTT   =   o   X 


2t  32 

D       2tSirX  3.375  X  d3 
P"  32  Xd 

=  0.662  X  S  X  t  X  d2  (150) 

Similarly,  if  the  pull  P  is  known,  the  pole-top  diameter  should  be : 


-4 


0.662  x  S  X  t 

EXAMPLE  OP  STRENGTH  CALCULATION 

Consider  a  pole  of  Eastern  white  cedar  designed  to  sustain 
a  pull  of  500  Ib.  applied  26  ft.  above  ground  level.  The  average 
breaking  stress  (from  table  of  constants)  is  4000  Ib.  per  square 
inch,  and  assuming  a  factor  of  safety  of  6  the  safe  working 
stress  is  S  =  660  Ib.  per  square  in.  The  other  numerical  values 
are: 

P  =  500  Ib. 
H  =  26  ft. 

<(from  table)  =  0.0165 
By  formula  (151) : 


Voi 


500 


/  0.662  X  660  X  0.0165 
=  8.33  in. 

dw  =  1.5  X  8.33  =  12.5  in. 

The  distance  below  point  of  application  of  load  of  the  section 
where  fiber  stress  is  a  maximum  is: 

x  =  £ -  =  252  in.  =  21  ft. 
at 


318  ELECTRIC  POWER  TRANSMISSION 

Therefore  this  pole,  if  subject  to  a  load  about  six  times  greater 
than  the  maximum  working  load,  may  be  expected  to  break 
26  -  21  =  5  ft.  above  ground  level. 

Double  pole  supports  of  the  type  illustrated  in  Fig.  114  will  be 
twice  as  strong  as  each  of  the  component  poles  in  resisting  stresses 
applied  in  the  direction  of  the  line;  but  they  will  be  able  to  with- 
stand about  five  times  as  great  a  load  as  the  single  pole  when  the 
stresses  are  in  a  direction  at  right  angles  to  the  direction  of 
the  line.  When  loaded  in  this  manner  up  to  the  breaking  point, 
these  double  poles  of  the  "A"  type  usually  fail  through  the 
buckling  of  the  member  in  compression  due  to  initial  want  of 
straightness.  The  strength  of  both  the  "A"  and  the  "H" 
type  of  pole  structure  can  to  some  extent  be  increased  by  judi- 
cious and  rigid  bracing. 

150.  Deflection  of  Wood  Poles. — It  is  now  generally  recognized 
that  there  are  advantages  in  having  transmission-line  supports 
with  flexible  or  elastic  properties.     The  ordinary  single  wood  pole 
is  very  elastic,  and  will  return  very  nearly  to  its  original  form  after 
having  been  deflected  considerably  by  abnormal  stresses.     The 
figures  given  for  the  elastic  modulus  in  the  table  previously  re- 
ferred to  are  subject  to  correction  for  different  qualities  and 
samples  of  the  same  timber.     It  is  well  to  make  a  few  experiments 
on  the  actual  poles  to  be  used  if  accuracy  in  calculated  result  is 
desired.     The  double-pole  structures  of  the  "  A  "  or  "  H  "  type  will 
have  about  half  the  deflection  of  the  single  poles  in  the  direction 
of  the  line,  and,  of  course,  very  much  less  in  a  direction  at  right 
angles  to  the  line.     An  "A"  pole  of  usual  construction  with  the 
two  poles  subtending  an  angle  of  6^  degrees  will  deflect  only 
about  one-fiftieth  of  the  amount  of  the  single-pole  deflection 
under  the  same  transverse  loading.     The  movement  is  usually 
dependent  upon  the  amount  of  slip  between  the  two  poles  at  top, 
which  again  depends  upon  the  angle  subtended  by  the  poles.1 
If  this  angle  is  as  much  as  10  degrees  there  will  be  practically  no 
likelihood  of  the  poles  slipping  at  the  top  joint;  but  this  large 
angle  is  unsightly,  and  probably  makes  a  less  economical  struc- 
ture than  the  more  usual  angle  of  about  6^  degrees. 

151.  Calculation  of  Pole  Deflections. — Assume  the  pole  to  be 
fixed  firmly  in  the  ground,  and  that  there  is  no  yielding  of  founda- 

1  Much  useful  information  on  the  behavior  of  "A"  poles  under  test  is  to 
be  found  in  Mr.  C.  Wade's  paper  read  before  the  Institution  of  Electrical 
Engineers  on  May  2,  1907. 


TRANSMISSION-LINE  SUPPORTS 


319 


tions.  The  load  P  being  applied  in  a  horizontal  direction  at  the 
top  end,  as  indicated  in  Fig.  117,  the  pole  may  be  considered  as 
a  simple  cantilever,  the  deflection  of  which,  if  the  section  were 
uniform  throughout  the  entire  length,  would  be: 

PHS 


3MI 

where  5  and  H  are  in  inches;  7  is  the  moment  of  inertia  of  the 
section,  and  M  is  Young's  modulus  (pounds  per  square  inch). 

For  a  circular  section  I  =  -^-  where  d  is  the  diameter  of  the 

(cylindrical)  pole  in  inches.     The  formula 
then  becomes: 


6™Md< 

If  P  is  evenly  distributed,  as  would  be  the 
case  with  a  uniform  wind  pressure  on  the 
pole  surface,  regardless  of  other  loads,  the 
deflection  would  be: 


"SMI 

but  it  is  best  to  consider  the  wind  pressure 
on  pole  surface  as  a  single  equivalent  load 
concentrated  at  pole-top  and  added  to  the 
load  due  to  wind  pressure  on  the  wires. 
When  estimating  the  probable  value  of 
this  equivalent  load,  it  should  be  remem- 
bered that  the  wind  pressure  is  not  evenly 
distributed  along  the  length  of  the  pole, 
since  the  wind  velocity  at  .ground  level  is 
comparatively  small  and  increases  with  the 
height  above  ground  surface. 

The  formula  (152)  assumes  a  constant  diameter  throughout 
length  of  pole,  and  the  question  therefore  arises  as  to  where  the 
measurement  of  diameter  should  be  made  on  an  actual  pole. 
Mr.  S.  M.  Powell  has  shown  that,  on  the  assumption  of  a  uni- 
form taper,  the  quantity  d4  in  formula  (152)  should  be  replaced 
by  (dgs  X  di)  where  dg  is  the  diameter  at  ground  level  and  di  is 
the  diameter  where  the  force  P  is  applied. 


FIG.    117. — Deflection 
of  wood  pole. 


320  ELECTRIC  POWER  TRANSMISSION 

EXAMPLE    OP  CALCULATION  OF   POLE-TOP   DEFLECTION 

Using  the  same  figures  as  in  the  example  of  strength  calcula- 
tions : 

P  =  500  Ib.  H  =  26  X  12  =  312  in. 

di  =  8.33  in.  t  =  0.0165 

d  =  8.33  +  (0.0165  X  312)  =  13.48  in. 
M  =  700,000 
then 

_  6.78  XPXH3 

'    M  (<43  X  di) 

6.78  X  500  X  (312)3 

700,000  X  (13.48)3  X  8.33 

=  7.2  in. 

When  possible  it  is  well  to  make  tests  on  a  few  actual  poles; 
then  for  similar  poles  of  the  same  material  subject  to  the  same 
loading: 

5cxd7x~Ti 

152.  Pole  Foundations. — A  permanent  deflection  of  the  pole 
when  the  stresses  are  abnormal  may  occur  owing  to  the  yielding 
of  the  earth  foundation;  but  this  is  unusual  if  the  poles  are 
properly  set  in  good  ground. 

The  diagram  Fig.  118  has  been  drawn  to  show  the  depth  to 
which  poles  of  various  heights  are  usually  set.  These  depths  are 
such  as  would  be  adopted  on  a  well-designed  pole  line,  and  need 
not  be  exceeded  except  in  special  cases.  In  marshy  or  otherwise 
unsatisfactory  ground,  special  means  must  be  adopted  to  provide 
a  reasonably  good  setting  for  the  pole  butts. 

Loam  and  gravel,  and  even  sand;  or  a  mixture  of  these,  pro- 
vides a  firm  foundation  for  poles.  A  pole  that  is  properly  set 
should  break  before  the  foundations  will  yield  to  any  appreciable 
extent.  Even  if  there  should  be  a  movement  of  the  pole  butt  in 
the  ground  with  excessive  horizontal  load  at  pole-top,  this  will 
result  in  a  firmer  packing  of  the  earth,  which  will  then  be  better 
fitted  to  resist  any  further  movement. 

Firm  sand,  gravel,  or  loam,  will  withstand  a  pressure  of  about 
4  tons  per  square  foot;  but  only  half  this  resistance  should  be 
reckoned  on  in  the  case  of  damp  sand,  moist  loam,  or  loose  gravel. 

Proper  supervision  is  necessary  to  ensure  that  the  earth  shall 


TRANSMISSION-LINE  SUPPORTS 


321 


be  packed  firmly  around  the  pole  when  refilling  the  holes.  This 
matter  of  tightly  packing  the  dirt  around  the  pole  butt  is  referred 
to  in  Appendix  II.  Although  no  attempt  has  been  made  to  treat 
adequately,  in  this  book,  of  practical  details  such  as  the  best 
method  of  digging  holes,  it  may  not  be  out  of  place  to  mention 
that  the  use  of  dynamite  for  digging  post  holes  appears  to  have 
met  with  success  where  the  method  has  been  carefully  studied 
and  intelligently  applied.  (Refer  to  the  Electrical  World,  June 
8,  1912  and  Feb.  7,  1914.) 


vur»c   A-  Poles  set  in  rock 

Carve   B-  Poles  set  in  solid  ground  on  straight 

Carve   C  -  Poles  set  in  solid  ground  at  corners 

or  in  poor  soil  on  straight  runs 
Carve    D-  Poles  set  at  corners  in  soft  ground 


25 


30  36  40  45  50  5 

Height  of  Pole  above  Ground  .Level  -  Feet 

FIG.   118.  —  Chart  giving  approximate  depth  of  holes  for  wood  poles. 


60 


153.  Spacing  of  Poles  at  Corners — Guy  Wires. — In  order  to 
reduce  the  stresses,  not  only  on  the  pole  itself,  but  also  on  the 
insulator  pins  and  cross-arms,  it  is  usual  to  shorten  up  the  spans 
on  each  side  of  the  corner  pole.  The  reduction  in  length  of  span 
will  depend  upon  the  amount  by  which  the  direction  of  the  wires 
departs  from  the  straight  run.  A  rough  and  ready  rule  is  to 
reduce  the  span  length  1^  per  cent,  for  each  degree  of  deviation 
from  the  straight  line.  For  angles  less  than  5  degrees,  it  is  not 
necessary  to  alter  the  span. 

It  is  not  advisable  to  turn  more  than  25  degrees  on  one  pole, 
and  whenever  the  side  strain  is  likely  to  be  excessive,  double 
cross-arms  and  insulators  should  be  used.  By  giving  proper 
attention  to  the  matter  of  guying  and  to  the  mechanical  con- 


322  ELECTRIC  POWER  TRANSMISSION 

struction  generally,  it  is  not  difficult  to  meet  all  requirements  at 
points  where  a  change  of  direction  occurs. 

A  safe  plan  is  to  assume  that  a  corner  pole  must  carry  the  full 
load  without  breaking  if  the  guy  wire  or  wires  should  fail  to  take 
their  proper  share  of  the  load:  but  all  corner  poles  should  be 
propped  or  guyed  for  extra  safety,  and  to  avoid  the  unsightly 
appearance  of  poles  bent  under  heavy  side  stresses  or  set  at  an 
angle  with  the  vertical. 

Sometimes  when  sharp  corners  have  to  be  turned,  the  spans  on 
each  side  are  "  dead-ended  "  on  poles  with  double  fixtures.  Such 
poles  are  head-guyed,  and  the  span  adjoining  the  guyed  pole  is 
usually  shortened,  being  not  more  than  three-fifths  of  the  average 
spacing.  For  further  particulars  of  common  practice  in  guying 
poles  in  special  positions,  the  reader  is  referred  to  the  sample 
specification  for  wooden  pole  line  in  Appendix  II. 


w 

FIG.   119.  —  Diagram  of  stresses  at  corner  pole. 

The  non-synchronous  swaying  of  wires  in  a  high  wind,  al- 
though uncommon,  sometimes  occurs  on  wood  pole  lines,  being 
aggravated  by  the  difference  in  the  natural  period  of  oscillation 
of  poles  and  wires.  This  trouble  can  generally  be  cured  by  guy- 
ing one  or  more  of  the  poles  at  the  place  where  the  wires  have 
been  found  to  swing  non-synchronously. 

Guy  wires  should  be  of  galvanized  stranded  steel  cable,  the 
breaking  strength  of  which  should  preferably  not  exceed  about 
34  tons  per  sq.  in.  The  reason  for  this  limitation  of  strength  is 
that  the  high-strength  steel  is  usually  too  hard  to  allow  of  proper 
handling  and  finishing  off. 

154.  Load  to  be  Carried  by  Corner  Poles.  —  If  P  is  the  total 
tension  in  pounds  of  all  the  wires  on  each  side  of  the  corner  pole, 
and  if  6  is  the  angle  of  deviation  as  indicated  in  Fig.  119,  then  the 
resultant  pull  in  the  direction  OW  at  the  pole  top  will  be, 


W  (Ib.)  =  2P  sin  (153) 

The  stress  in  the  guy  wire  is  readily  calculated  when  the  angle 


TRANSMISSION  LINE  SUPPORTS 


323 


a  (Fig.  120)  which  the  wire  makes  with  the  vertical  is  known. 
If  W  is  the  side  pull  as  calculated  by  formula  (153),  then 

W 


Tension  in  guy  wire  = 


CD 


Q54) 


155.  Props  or  Struts  —  Wood  Poles  in  Compression.  —  Some- 
times it  is  difficult  or  impossible  to  provide  guy  wires  in  certain 
locations;  or  impurities  in  the  atmosphere  may  render  the 
use  of  props  or  push  braces  preferable  to  guy  wires.  In  such  cases 
it  is  necessary  to  know  approximately  what  load  a  wooden  pole 
will  support  in  compression,  that  is  to  say  when  used  or  considered 
as  a  column.  Instead  of  using  the  values  of  unit  stress,  S,  as 


c  D 

FIG.  120. — Diagram  of  stress  in  guy  wire. 

given  in  the  Table  on  p.  314,  the  ultimate  stress  which  a  wood 
column  will  stand  in  compression  should  be  calculated  by  the 
empirical  formula: 

Stress  in  compression  (Ib.  per  sq.  in.)  =  S  (1  —  ^^j      (155) 

where  I  is  the  length  in  inches,  and  d  is  the  diameter  or  least 
thickness  at  the  center  of  the  strut. 

Example. — Calculate  the  safe  load  for  a  prop  of  Douglas  fir, 
8  in.  diameter  and  10  feet  long,  assuming  a  safety  factor  of  6. 

By  formula  (155)  the  breaking  stress  will  be 

6500  (l  -  nn  _  J) 


and  the  maximum  safe  load  will  be 

~-  X  ~  X  64  =  41,000  Ib. 


324  ELECTRIC  POWER  TRANSMISSION 

156.  Reinforced  Concrete  Poles. — As  substitutes  for  wood 
poles  supporting  overhead  wires,  steel  poles  of  the  tubular  form 
and  latticed  steel  masts  are  used.  The  full  advantage  of  the 
galvanized  or  painted  steel  structure  is  best  realized  in  the 
high  towers  with  extra  wide  spacing,  such  as  are  used  for  the 
transmission  of  electric  energy  at  high  pressures.  The  use  of 
Portland  cement  for  moulded  poles  of  moderate  height  is  by  no 
means  new;  the  experimental  stage  has  long  ago  been  passed,  and 
with  the  deplorable  but  no  less  rapid  depletion  of  our  forests  and 
the  incomparably  longer  life  of  the  concrete  poles,  these  will 
probably  be  used  in  increasing  numbers  during  the  next  few 
years. 

There  is  much  to  be  said  in  favor  of  the  wood  pole  when  the 
right  kind  of  timber,  properly  seasoned  and  treated,  is  used; 
but,  apart  from  the  general  unsightliness  of  wood  poles  in  urban 
districts,  their  life  is  uncertain  and  always  comparatively  short. 
In  Switzerland  the  experiment  has  been  tried  of  covering  the 
ordinary  wood  pole  with  concrete  mortar  about  1  in.  thick.  The 
strength,  and  especially  the  life,  are  greatly  increased  thereby, 
as  the  decay  which  so  frequently  occurs  at  ground  level  will  be 
largely,  if  not  entirely,  prevented;  but  it  is  doubtful  whether  the 
system  will,  in  the  long  run,  prove  satisfactory  or  economical. 
The  ideal  material  to  use  for  reinforcing  concrete  is  undoubtedly 
steel  or  iron.  Longitudinal  rods  or  bars  of  iron  can  be  placed 
exactly  where  required  to  strengthen  those  parts  of  the  pole  sec- 
tion that  will  be  in  tension,  and  the  concrete,  filling  up  the  spaces 
between  the  reinforcing  rods,  takes  the  place  of  all  bracing  and 
stiffening  members  of  the  ordinary  steel  structure  in  an  almost 
perfect  manner.  It  is  probably  at  this  time  generally  admitted 
that  iron  embedded  in  cement  will  last  almost  indefinitely  without 
suffering  any  deterioration.  When  excavating  for  the  founda- 
tions of  the  new  General  Post  Office  in  London,  England,  some 
old  Roman  brickwork  was  discovered  in  which  the  hoop-iron 
bonds  were  still  bright  and  in  perfect  condition.  The  life  of  a 
concrete  pole  is,  in  fact,  almost  unlimited,  a  consideration  which 
should  riot  be  overlooked  when  estimating  the  relative  costs  of 
different  kinds  of  supporting  structures.  It  requires  no  painting 
and  practically  no  attention  once  it  is  erected.  If  any  small 
cracks  should  at  any  time  develop,  they  can  readily  be  filled  with 
cement. 

While  referring  to  the  advantages  of  the  cement  pole  it  may  be 


TRANSMISSION-LINE  SUPPORTS  325 

added  that  every  pole  is  virtually  a  lightning  rod,  an  advantage 
which  it  shares  with  the  steel  pole  or  tower.  On  lines  where  both 
timber  and  concrete  poles  have  been  used  and  where  many  wood 
poles  have  been  shattered  by  lightning,  the  concrete  poles  have 
rarely  been  struck.  There  is  an  instance  of  a  concrete  pole  of  the 
Marseilles  (111.)  Land  &  Water  Company  having  been  struck,  but 
the  only  damage  done  was  the  chipping  out  of  a  small  piece  at  the 
top  of  the  pole  and  one  at  the  bottom  where  the  current  entered 
the  ground  after  following  down  the  steel  reinforcing  bars  inside 
the  pole. 

157.  Weight  and  Cost  of  Concrete  Poles.— The  weight  of 
concrete  poles  is  necessarily  considerable,  and  unless  the  poles 
are  made  near  the  site  where  they  will  be  erected  the  cost  of 
transportation  will  generally  be  prohibitive.  Concrete  poles 
usually  measure  6  in.  at  the  top  with  a  base  width  of  10  in.  to 
14  in.  depending  on  the  height.  It  is,  however,  quite  permissible 
to  use  poles  with  5  in.  top  measurement,  in  which  case  the  base 
measurement  might  be  about  9  in.  for  poles  not  exceeding 
25  to  30  feet  overall  length. 

The  weight  of  concrete  is  about  150  Ib.  per  cubic  foot,  and  the 
cost  of  poles  will  range  between  35c.  and  70c.  per  100  Ib. 
Should  it  be  found  that  conditions  of  labor,  transportation,  etc., 
are  such  that  the  cost  would  be  in  excess  of  70c.  per  100  Ib.  it  is 
probable  that  steel  or  wood  supports  would  prove  more  econom- 
ical than  reinforced  concrete. 

On  the  basis  of  50c.  per  100  Ib.  weight  of  the  finished  pole, 
the  following  figures  indicate  roughly  the  approximate  cost  of 
concrete  poles.  The  lengths  given  are  the  overall  lengths, 
including  the  portion  buried  in  the  ground.  The  weights  are 
such  as  might  be  expected  for  poles  designed  with  hollow  cores. 

Length,  ft.  Weight,  Ib.  Cost 


30 

1800 

$9.00 

35 

2200 

11.00 

40 

3600 

16.00 

The  great  weight  of  concrete  poles  is  probably  the  most  serious 
objection  to  their  more  general  adoption  in  the  place  of  wood 
poles,  where  the  latter  are  not  readily  obtainable  or  where  their 
appearance  is  unsightly. 


326  ELECTRIC  POWER  TRANSMISSION 

It  is  probable  that  the  concrete  poles  of  cross-country  transmis- 
sion lines  are  usually  made  somewhat  heavier  than  the  strength 
requirements  necessitate  because,  being  moulded  on  site,  not 
always  with  the  best  and  most  convenient  appliances,  they  are 
made  solid  throughout  or  through  a  large  part  of  their  length, 
whereas  a  hollow  construction  would  have  been  adopted  had 
suitable  collapsible  cores  been  available. 

Poles  up  to  35  ft.  in  length  are  usually  moulded  in  a  horizontal 
position,  the  forms  being  removed  after  three  or  four  days.  After 
a  period  of  seasoning  lasting  from  two  to  three  weeks  they  are 
erected  in  the  same  manner  as  wood  poles. 

Poles  longer  than  35  ft.  are  often  moulded  in  a  vertical  position. 
The  forms  are  set  up  immediately  over  the  hole  previously  pre- 
pared for  the  pole  base.  They  are  set  truly  vertical  and  tempo- 
rarily guyed,  the  reinforcing  inside  the  form  being  held  together 
and  in  position  by  whatever  means  of  tying  or  bracing  may  be 
adopted.  Sometimes  iron  wire  is  used,  but  more  uniform  results 
are  obtained  by  using  specially  designed  iron  distance  pieces  with 
the  required  spacing  between  them.  The  concrete  is  raised  to 
the  top  of  the  mould  by  any  suitable  and  economic  means  (pref- 
erably direct  from  the  concrete  mixer  by  an  arrangement  equiva- 
lent to  the  ordinary  grain  elevator)  and  is  dropped  in.  By 
this  means  the  hole  in  the  ground  is  entirely  filled  with  concrete. 
No  tamping  is  required,  a  firm  hold  being  obtained,  since  the 
ground  immediately  surrounding  the  concrete  base  has  not  been 
disturbed. 

The  best  quality  of  crushed  stone  and  sand  should  be  used,  the 
usual  proportions  being:  cement,  one  part;  sand,  two  parts; 
crushed  stone,  three  or  four  parts,  not  too  large  to  pass  through  a 
%-in.  screen.  The  mixture  used  for  the  poles  on  the  Pennsyl- 
vania Railroad  is  1.5:2:4.  When  gravel  is  used  the  mixture 
may  be  one  part  of  Portland  cement  to  five  parts  of  gravel,  pro- 
'vided  that  the  latter  is  graded,  including  sand,  and  with  the 
largest  pieces  of  a  size  to  pass  through  a  %-in.  screen. 

The  cost  of  concrete  poles,  when  the  long  life  and  other  ad- 
vantages are  taken  into  account,  does  not  compare  unfavorably 
with  that  of  other  types;  but  it  must  not  be  overlooked  that  the 
cost  of  materials  and  labor  required  to  manufacture  the  poles  does 
not  represent  the  cost  of  the  finished  pole  erected  in  position. 
Much  valuable  information  on  the  costs  of  manufacture  and  hand- 
ling of  concrete  poles,  together  with  practical  details  relating  to 


TRANSMISSION-LINE  SUPPORTS  327 

methods  of  manufacture,  will  be  found  in  Mr.  R.  A.  Lundquist's 
book  on  Transmission  Line  Construction.1  The  reader  is  also 
referred  to  an  article  by  Mr.  J.  G.  Jackson  who  describes  in  the 
Electrical  World  of  Jan.  17,  1914,  how  the  concrete  poles  used  on 
the  Toronto  Hydro-electric  system  were  manufactured.  The 
article  by  Mr.  R.  D.  Coombs,  in  the  Electrical  World  of  Feb.  6, 
1915,  and  a  more  recent  article  entitled  "Concrete  poles  carry 
22,000-volt  power  line,"  in  the  Electrical  World  of  Feb.  9,  1918 
(Vol.  71,  p.  296)  should  also  be  consulted  by  those  desiring  fur- 
ther information  on  concrete-pole  transmission  lines. 

As  an  example  of  a  concrete-pole  line,  the  transmission  line 
of  the  Northern  Illinois  Light  and  Traction  Company,  of  Mar- 
seilles, 111.,  may  be  mentioned.  This  company  transmits  three- 
phase  energy  at  from  30,000  volts  to  33,000  volts.  Most  of  the 
poles  used  are  about  30  ft.  high,  spaced  from  125  ft.  to  132  ft. 
apart.  The  section  is  square,  with  6-in.  sides  at  the  top  of  the 
pole  and  9  in.  at  the  base.  The  reinforcing  consists  of  six  ^-in.- 
square  steel  bars  through  the  entire  length  of  the  pole.  Many 
of  the  concrete  poles  on  this  line  have  now  been  in  position  over 
nine  years,  and  they  have  given  entire  satisfaction. 

158.  Strength  and  Stiffness  of  Concrete  Poles. — When  design- 
ing a  concrete  pole  to  withstand  a  definite  maximum  horizontal 
load  applied  near  the  top,  the  pole  is  treated  as  a  beam  fixed  at 
one  end  and  loaded  at  the  other.  The  calculations  are  very 
simple  if  certain  assumptions  are  made,  these  being  as  follows: 

(1)  Every  plane  section  remains  a  plane  section  after  bending. 

(2)  The  tension  is  taken  by  the  reinforcing  rods. 

(3)  The  concrete  adheres  perfectly  to  the  steel  rods. 

(4)  The  modulus  of  elasticity  of  concrete  is  constant  within 
the  usual  limits  of  stress. 

The  ultimate  crushing  stress  of  the  concrete  may  be  taken  at 
from  2000  to  2600  Ib.  per  square  inch.  The  reinforcing  bars 
should  be  covered  with  concrete  to  a  depth  of  not  less  than  1  in. 
The  effect  of  keeping  the  reinforcing  bars  under  tension  while  the 
concrete  is  poured  in  the  mould  and  until  it  has  hardened  suffi- 
ciently to  support  the  strain  itself  has  been  tried  and  found  to  im- 
prove the  performance  of  the  poles,  but  it  is  doubtful  whether  the 
extra  apparatus  and  labor  required  are  justifiable  on  economic 
grounds.  When  subjected  to  excessive  load  a  concrete  pole  will 
generally  yield  by  the  crushing  of  the  material  in  the  base  near 

1  McGraw-Hill  Book  Co. 


328 


ELECTRIC  POWER  TRANSMISSION 


ground  level;  but,  unless  it  is  pulled  out  of  its  foundations,  it 
will  not  fall  to  the  ground. 

The  comparative  rigidity  of  concrete  poles  cannot  be  said 
to  be  a  point  in  their  favor,  as  the  flexibility  and  elasticity  of 
wood  poles  and  some  forms  of  steel  structures  are  features  of 
undoubted  advantage  under  certain  conditions.  On  the  other 
hand,  the  degree  of  deflection  of  concrete  poles  before  breaking 
is  remarkable.  The  elastic  limit  is  variable,  and  no  exact  figure 
can  be  given  for  the  elastic  modulus  of  cement  concrete;  it  may 
be  as  low  as  1,000,000  but  for  a 
1:2:4  mixture  2,000,000  may  be 
taken  as  a  good  average  figure  for 
approximate  calculations. 

Some  tests  made  on  30-ft.  con- 
crete poles  gave  deflections  of  from 
3  in.  to  4  in.  at  a  point  near  the  top 
of  pole,  when  subjected  to  a  test  load 
equal  to  about  double  the  maximum 
working  load.1  Another  series  of 
tests  made  recently  in  England  on 
some  44-ft.  poles  of  hollow  section, 
17  in.  square  at  the  base  and  8  in. 
square  at  the  top  (inside  dimensions 
13  in.  and  4  in.  respectively),  with 
loads  applied  38.5  ft.  above  ground 
level,  gave  a  deflection  of  66  in. 
under  a  horizontal  load  of  10,500  lb., 
and  the  permanent  set  on  removal 
of  load  was  21  in.  The  pole  did  not 
fail  completely  until  the  deflection 
was  78  in. 

The  illustration  Fig.  121  shows  a  typical  concrete  pole  of 
hollow  section  suitable  for  carrying  six  transmission  wires  on  two 
wooden  cross-arms.  The  pole  is  35  ft.  long  over  all,  about  6  ft. 
being  buried  in  the  ground.  With  a  top  measurement  of  7  in. 
square  and  a  taper  to  give  an  increase  of  1  in.  width  for  every  5  ft. 
of  length,  the  size  at  the  bottom  will  be  14  in.  square.  The 

1  These  poles  were  probably  of  large  cross-section.  Some  tests  made  on 
poles  measuring  10  in.  square  at  the  base  and  32  ft.  high  gave  a  deflection 
of  just  over  2  ft.  with  a  horizontal  load  of  2000  lb.  applied  near  the  top. 


Section  A-A 


Fia.  121. — Concrete  pole  of 
hollow  section. 


TRANSMISSION-LINE  SUPPORTS  329 

drawing  shows  a  section  through  the  hollow  pole  taken  at  a 
point  about  4  ft.  above  the  ground  level.  Iron  spacing  pieces,  as 
here  shown,  or  their  equivalent,  must  be  placed  at  intervals  to 
hold  the  longitudinal  steel  reinforcing  bars  in  the  proper  position. 
The  number  of  rods  will  vary  with  the  distance  below  the  point 
of  application  of  the  load.  The  bending  moment  to  be  resisted 
at  every  point  being  known  and  the  taper  of  the  pole  decided 
upon,  the  amount  of  reinforcing  required  at  any  given  section  is 
easily  calculated.  The  weight  of  a  pole  as  illustrated  would  be 
about  2700  Ib.  without  fixtures.  The  reinforcing  rods  and 
spacing  rings  would  account  for  approximately  one-seventh  of 
the  total  weight.  A  factor  of  safety  of  four  is  generally  employed 
in  strength  calculations  of  reinforced  concrete  poles.  In  some 
cases  the  calculations  have  been  based  on  a  safety  factor 
of  5;  but  there  appears  to  be  no  justification  for  using  so  large 
a  factor. 

159.  Steel  Poles  and  Towers — Introductory  Remarks. — It  can- 
not be  said  that  there  is  at  the  present  tune  one  standard  type  of 
steel  structure  for  supporting  the  conductors  of  overhead  trans- 
mission lines;  neither  is  it  likely  that  one  particular  design 
will  ever  be  found  suitable  for  all  countries,  climates  and  voltages. 
Any  kind  of  supporting  structure  which  will  economically  fulfil 
the  necessary  requirements  will  answer  the  purpose  of  the  trans- 
mission line  engineer,  who  merely  requires  a  durable  mechanical 
structure  to  carry  a  variable  number  of  insulators  at  a  height 
above  ground,  and  with  a  spacing  between  them,  depending  upon 
the  voltage  of  transmission  and  the  length  of  span. 

As  a  substitute  for  wood  poles,  steel  tubes  have  been  used, 
either  in  one  piece,  or  built  up  of  several  sections  of  different 
sizes  in  order  to  economize  material  and  give  a  large  diameter  at 
the  bottom  where  the  bending  moment  is  greatest,  and  a  small 
diameter  at  the  top  where  the  bending  moment  is  negligible. 
Steel  poles  of  considerable  height,  suitable  for  longer  spans,  may 
be  built  up  of  three  or  four  vertical  tubes  of  comparatively  small 
diameter  joined  and  braced  together  at  suitable  intervals  to  give 
stiffness  to  the  structure.  It  is  doubtful  whether,  in  the  long 
run,  such  composite  tubular  structures  will  hold  their  own  against 
the  small-base  latticed  steel  masts  built  up  of  standard  sections  of 
rolled  steel,  as  used  extensively  on  the  continent  of  Europe,  and, 
to  a  relatively  smaller  extent,  in  America.  The  term  "tower" 
is  applied  mainly  to  the  light  steel  structures  in  which  the  spac- 


330  ELECTRIC  POWER  TRANSMISSION 

ing  between  the  main  upright  members,  at  ground  level,  is  large 
compared  with  the  height  of  the  structure;  the  usual  proportion 
— which  will  generally  be  found  to  be  the  most  economical  in 
material — being  1  to  4;  that  is  to  say,  if  the  base  is  square,  the 
side  of  this  square  will  be  about  one-quarter  of  the  distance  from 
the  point  of  measurement  to  the  top  of  the  tower.  If  the  towers 
are  large,  the  footings  are  usually  separate  pieces  which  are  cor- 
rectly set  in  the  ground  by  means  of  a  templet,  and  to  which  the 
legs  of  the  tower  proper  are  afterward  bolted.  A  good  example 
of  large  steel  towers  is  to  be  found  in  the  100,000-volt  trans- 
mission line  of  the  Great  Western  Power  Co.  of  California.  Two 
three-phase  circuits  are  carried  on  these  towers,  the  vertical 
spacing  between  the  cross-arms  being  10  ft.  There  are  three 
cross-arms,  each  carrying  two  conductors — one  at  each  end. 
The  horizontal  spacing  between  wires  is  17  ft.  on  the  two 
upper  cross-arms  and  18  ft.  on  the  lower  cross-arm,  which  is  51 
ft.  above  ground  level.  No  conductor  is  closer  than  6  ft. 
5  in.  to  the  steel  structures,  this  being  the  minimum  clearance 
in  the  horizontal  direction.  The  average  distance  between 
towers  is  750  ft.,  and  they  are  joined  at  the  top  by  a  grounded 
guard  wire  5  ft.  above  the  bottom  of  the  highest  cross-arm. 
The  base  of  the  tower  measures  17  ft.  square,  the  parts  under 
ground  being  separate  pieces  of  steel,  buried  to  a  depth  of  6  ft. 
to  which  the  tower  proper  is  bolted  after  being  assembled  and 
erected  on  site. 

Although  the  larger  towers  are  nearly  all  built  of  the  square 
type  as  used  for  windmills,  there  is  a  notable  exception  in  the 
case  of  the  140,000- volt  line  in  Michigan,  where  the  towers 
are  of  a  special  three-legged  type,  built  up  entirely  of  angle 
sections. 

Fig.  122  shows  a  typical  form  of  small-base  latticed  steel  mast 
on  the  transmission  lines  of  the  Iowa  Railway  and  Light  Com- 
pany, Cedar  Rapids,  Iowa;  while  Fig.  123  is  a  good  example  of 
square  base  tower  carrying  two  three-phase  lines.  These 
illustrations  are  reproduced  from  photographs  kindly  supplied 
by  the  Ohio  Brass  Co.  of  Mansfield,  Ohio.  The  large  towers  of 
Fig.  123  were  designed  and  constructed  by  the  American  Bridge 
Co.  of  Pittsburg  for  the  American  Gas  and  Electric  Company's 
130,000-volt  transmission  between  Wheeling,  W.  Va.  and  Canton, 
Ohio.  The  six  conductors  are  each  of  200,000  circular  mil 
cross-section,  and  the  two  grounded  guard  wires  are  of  the  same 


TRANSMISSION-LINE  SUPPORTS  331 

size.  The  line  is  55  miles  long,  and  the  average  length  of  span  is 
580  feet. 

The  economical  span  for  the  square  latticed  poles,  of  the  type 
shown  in  Fig.  122,  is  probably  something  less  than  450  feet; 
but  for  comparatively  light  lines,  this  form  of  structure  with 
spans  of  400  to  430  feet  is  very  satisfactory. 

160.  Flexible  Towers. — Although  calculations  of  stresses  in 
transmission  lines  are  usually  based  on  the  assumption  that  the 
ends  of  each  span  are  firmly  secured  to  rigid  supports;  this  con- 
dition is  rarely  fulfilled  in  practice;  there  is  some  "give"  about 
the  poles  or  towers,  especially  when  the  line  is  not  absolutely 
straight,  and  the  insulator  pins  will  bend  slightly  and  relieve  the 
stress  when  this  tends  to  reach  the  point  at  which  the  elastic 
elongation  of  the  wires  will  be  exceeded.  Then,  again,  the  wires 
will  usually  slip  in  the  ties  at  the  insulators,  even  if  these  ties  are 
not  specially  designed  to  yield  or  break  before  damage  is  done  to 
the  insulators  or  supporting  structures.  The  use  of  the  suspen- 
sion type  of  insulator,  which  is  now  becoming  customary  for 
the  higher  voltages,  adds  considerably  to  the  flexibility  of  the 
line. 

In  regard  to  the  towers  themselves,  all  steel  structures  for 
dead-ending  lines  or  sections  of  lines  are  necessarily  rigid,  and 
the  usual  light  windmill  type  of  tower  with  wide  base  is  also 
without  any  appreciable  flexibility.  The  latticed  steel  masts,  as 
used  more  generally  in  Europe  than  in  America,  are  slightly 
more  flexible,  and  the  elastic  properties  of  the  ordinary  wood 
pole  are  well  known.  The  deflection  of  a  wood  pole  may  be 
considerable,  and  yet  the  pole  will  resume  its  normal  shape  when 
the  extra  stress  is  removed.  There  is  much  to  be  said  in  favor 
of  so-called  flexible  steel  structures;  that  is  to  say,  of  steel  sup- 
ports designed  to  have  flexibility  in  the  direction  of  the  line, 
without  great  strength  to  resist  stresses  in  this  direction;  but 
with  the  requisite  strength  in  a  direction  normal  to  the  line,  to 
resist  the  side  stresses  due  to  wind  pressures  on  the  wires  and  the 
supports  themselves. 

Such  a  design  of  support  has  the  important  advantage  of  being 
cheaper  than  the  rigid  tower  construction,  in  addition  to  which  it 
gives  flexibility  where  this  is  advantageous,  with  the  necessary 
strength  and  stiffness  where  required.  The  economy  is  not  only 
in  the  cost  of  the  tower  itself  but  in  the  greater  ease  of  transport 
over  rough  country,  the  preparation  of  the  ground,  and  erection. 


332  ELECTRIC  POWf/R  TRANSMISSION 

The  advantages  of  flexibility  in  the  direction  of  the  line  are 
considerable.  Probably  the  most  severe  stresses  which  a  trans- 
mission line  should  be  capable  of  withstanding  are  those  due  to 
the  breakages  of  wires.  Such  breakages  may  be  caused  by  ab- 
normal wind  pressures,  by  trees  falling  across  the  line,  or  by  a 
burn-out  due  to  any  cause.  Suddenly  applied  stresses  such  as  are 
caused  by  the  breaking  of  some  or  all  of  the  wires  in  one  span  are 
best  met  by  being  absorbed  gradually  into  a  flexible  system. 
The  supports  on  each  side  of  the  wrecked  span  will  bend  toward 
the  adjoining  spans  because  the  combined  pull  of  all  the  wires 
in  the  adjoining  spans  is  greater  than  the  pull  of  the  remaining 
wires,  if  any,  in  the  wrecked  span.  This  movement  of  the  pole 
top  results  in  a  reduction  of  tension  in  the  wires  of  the  adjoin- 
ing span  owing  to  the  increased  sag  of  these  wires;  there  will  be 
an  appreciable  deflection  of  the  second  and  third  poles  beyond  the 
break,  but  the  amount  of  these  successive  deflections  will  de- 
crease at  a  very  rapid  rate  and  will  rarely  be  noticeable  beyond 
the  fourth  or  fifth  pole.  It  is  obvious  that,  as  the  remaining 
wires  in  the  faulty  span  tighten  up,  the  stress  increases;  but 
the  combined  pull  of  these  wires  on  the  pole  top  is  smaller  than 
it  was  before  the  accident,  since  it  is  assisted  by  the  pull  of  the 
deflected  poles,  and  these  joint  forces  are  balanced  by  the 
combined  pull  of  all  the  wires  in  the  adjoining  sound  span,  which 
pull,  as  previously  mentioned,  is  smaller  than  it  was  under  normal 
conditions. 

The  greater  the  flexibility  of  the  supports  in  the  direction  of  the 
line,  the  smaller  will  be  the  extra  load  which  any  one  support 
will  be  called  upon  to  withstand;  on  the  other  hand,  it  is  usual  to 
provide  anchoring  towers  of  rigid  design  about  every  mile  on 
straight  runs,  and  also  at  angles,  in  addition  to  which  every 
fifth  or  sixth  flexible  tower  may  be  head-guyed  in  both  directions. 
In  the  writer's  opinion,  too  much  stress  is  usually  laid  on  the 
necessity  for  providing  rigid  strain  towers  at  frequent  in- 
tervals to  prevent  the  effect  of  a  break  in  the  wires,  or  the  failure 
of  a  single  support  travelling  along  the  line  and  causing  in- 
jury to  an  indefinite  number  of  consecutive  spans.  The  semi- 
flexible  structures  referred  to  are  not  designed,  or  should  not 
be  designed,  without  very  careful  consideration  of  the  conditions 
they  have  to  fulfil;  and  there  appear  to  be  no  scientific  reasons, 
and  no  records  of  injury  to  actual  lines,  which  would  justify 
the  assumption  that  transmission  lines  of  this  type  are  liable  to 


FIG.  123. — Typical  steel  tower  transmission  line. 


(Facing  page  332) 


FIG.  124.— Flexible  steel  tower  line. 


TRANSMISSION-LINE  SUPPORTS  333 

be  wrecked  in  the  same  wholesale  or  cumulative  manner  as  a 
row  of  card  houses.  The  strain  towers  are  undoubtedly  helpful 
at  the  time  when  the  wires  are  strung;  but  it  is  possible  that 
they  are  used  at  more  frequent  intervals  than  the  economies  of 
sound  engineering  require. 

In  level  country,  a  modified  clamp  in  the  form  of  a  sleeve 
with  flared  ends  may  be  used  in  conjunction  with  the  lighter 
(and  cheaper)  flexible  type  of  supporting  structures;  and  a  com- 
promise between  the  loose  sleeve  and  the  rigid  clamp  or  tie  can 
be  used  on  all  lines  with  flexible  supports.  Clamps  of  this 
type  are  designed  to  allow  the  wire  to  slip  before  the  combined 
pull  of  all  the  wires  exceeds  the  load  that  will  permanently  de- 
form the  supporting  structure;  and  although  it  is  almost  impos- 
sible to  ensure  that  such  devices  will  remain  for  any  length  of 
time  in  the  same  condition  as  when  they  are  installed,  yet 
they  will  generally  afford  a  reasonable  degree  of  protection  in 
the  event  of  the  simultaneous  breaking  of  all  the  wires  in  one 
span.  It  is  not  unusual  to  carry  a  galvanized  Siemens-Martin 
steel  strand  cable  above  the  high-tension  conductors  on  the  tops 
of  the  steel  structures.  This  has  the  double  advantage  of 
securely,  but  not  rigidly,  tying  together  the  supports,  and  of 
providing  considerable  protection  against  the  effects  of  light- 
ning. The  disadvantages  are  increased  cost  and  possible — but 
not  probable — danger  of  the  gounded  wire  falling  on  to  the  con- 
ductors and  causing  interruption  of  supply. 

The  dead-end  towers  should  be  capable  of  withstanding  the 
combined  pull  of  all  the  wires  on  one  side  only,  when  these  are 
loaded  to  the  expected  maximum  limit,  without  the  foundations 
yielding  or  the  structure  being  stressed  beyond  the  elastic  limit. 
The  flexible  supports  must  withstand,  with  a  reasonable  factor  of 
safety,  the  dead  weight  of  conductors,  etc.,  and  the  expected  maxi- 
mum side  pressures;  but  in  the  direction  of  the  line  their  strength 
must  necessarily  be  small,  otherwise  the  condition  of  flexibility 
cannot  be  satisfied. 

It  is  easy  to  design  braced  A-frame  or  H-frame  steel  structures 
of  sufficient  strength  to  withstand  the  dead  load  and  lateral 
pressure  and  yet  have  great  flexibility,  with  correspondingly 
reduced  strength,  in  the  direction  of  the  line.  Great  care  must 
be  used  in  designing  a  line  of  this  type  so  that  strength  and  dura- 
bility shall  not  be  sacrificed  to  lightness  and  flexibility  without 
very  carefully  considering  the  problem  in  all  its  aspects. 


334  ELECTRIC  POWER  TRANSMISSION 

The  assumptions  made  for  the  purpose  of  simplifying  strength 
calculations  are  not  always  permissible.  For  instance  the  effect 
of  a  twist  in  these  flexible  structures  is  sometimes  overlooked; 
but  when  there  is  inequality  of  tension  in  the  wires  on  the  two 
sides  of  the  structure,  the  fact  that  the  section  passing  through 
the  two  main  upright  members  is  no  longer  a  plane  at  right 
angles  to  the  direction  of  the  line  accounts  for  the  lessened 
strength  of  such  a  flexible  design  to  resist  loads  due  to  high  winds 
blowing  across  the  line.  It  is  not  safe  to  adopt  the  flexible  type 
of  transmission  line  support  without  expert  advice  and  adequate 
engineering  supervision.  As  an  approximate  indication  of  pres- 
ent-day practice  in  arriving  at  the  load  in  the  direction  of  the 
line  for  which  flexible  structures  should  be  designed,  it  may  be 
stated  that  a  load  of  from  one-twentieth  to  one-tenth  of  the  total 
load  for  which  the  rigid-strain  towers  are  designed  should  not 
stress  the  intermediate  flexible  structures  beyond  the  elastic 
limit.  It  is  well  to  bear  in  mind  that  at  the  moment  of  rupture  of 
one  or  more  wires  on  a  "flexible"  transmission  line  the  resulting 
stresses  in  the  structures  and  remaining  wires  will  be  in  the  nature 
of  waves  or  surges  until  the  new  condition  of  equilibrium  is 
attained,  and  the  maximum  stresses  immediately  following  a 
rupture  will  generally  exceed  the  final  value. 

The  mathematics  required  for  the  exact  determination  of 
stresses  and  deflections  in  a  transmission  line  consisting  of  a 
series  of  flexible  poles  is  of  a  very  high  order,  even  when  many 
assumptions  are  made  which  practical  conditions  may  not  justify; 
but  the  limiting  steady  values  of  these  stresses  and  deflections 
can  be  calculated  in  the  manner  described  in  Article  170  at  the 
end  of  this  chapter,  and  as  the  range  between  these  limits  will 
usually  be  very  small,  the  probable  maximum  stresses  under 
given  conditions  can  be  estimated  with  a  reasonable  degree  of 
accuracy.  The  illustration  Fig.  124  kindly  supplied  by  Messrs. 
Archbold  Brady  and  Co.,  shows  a  common  form  of  "flexible" 
high  voltage  transmission  line  following  a  railway. 

161.  Steel  Poles  for  Small  Short-distance  Transmission 
Schemes. — As  a  substitute  for  wood  poles,  light  steel  structures 
that  can  be  shipped  and  erected  in  one  piece  appear  to  be  gain- 
ing favor.  Small  amounts  of  energy  at  comparatively  low 
voltages  can  be  transmitted  over  distances  of  20  to  30  miles  by 
overhead  wires  supported  on  steel  poles  at  a  cost  which  need  be 
no  higher,  and  is  sometimes  even  lower,  than  if  the  less  durable 


FIG.  125. — Bates  one- 
piece  expanded  steel 
transmission  pole. 


FIG.  126. — Type  of  steel  pole 
manufactured  by  the  Carbo  Cor- 
poration. 


(Facina  page  334) 


TRANSMISSION-LINE  SUPPORTS  335 

and  less  sturdy  wood-pole  construction  is  adopted.  One  type 
of  steel  pole  for  small  lines  is  the  Bates  One  Piece  Expanded 
Steel  Truss  of  which  Fig.  125  is  an  example.  These  poles  are 
manufactured  by  the  Bates  Expanded  Steel  Truss  Co.  of  Chi- 
cago, with  the  idea  that  wooden  poles  are  gradually  giving  out 
and  that  this  is  a  practical  substitute,  the  pole  being  made  in 
one  piece  without  bolted  or  riveted  lattice  work.  A  pole  of  this 
type  is  easily  and  economically  painted.  Another  make  of  light- 
weight steel  pole  is  shown  in  Fig.  126.  This  is  manufactured  by 
the  Carbo  Corporation  of  Chicago,  which  makes  a  specialty  of 
steel  poles  for  the  economical  construction  of  moderate  voltage 
small-power  transmission  lines.  Steel  poles  of  the  type  under 
discussion  would  range  from  25  ft.  to  35  ft.  in  height  and  would 
be  spaced  from  250  to  300  feet  apart. 

162.  Loads  to  be  Resisted  by  Towers. — The  maximum  load 
which  a  tower  must  be  designed  to  withstand  will  depend  upon 
the  number  and  size  of  wires  to  be  carried  and  the  estimated 
ice  coating  and  wind  velocity.  Apart  from  the  wind  pressure  on 
the  structure  itself,  the  loading  in  a  direction  transverse  to  the 
line  will  be  equal  to  the  resultant  wind  pressure  on  all  the  wires 
(which  may  or  may  not  be  ice  coated,  depending  on  the  climate) ; 
the  effective  length  of  each  wire  being  the  distance  between 
supports. 

In  the  direction  of  the  line,  the  forces  are  normally  very  nearly 
balanced,  but  in  the  event  of  one  or  more  wires  breaking,  the 
unbalanced  load  may  be  considerable,  and  it  is  well  to  design 
the  towers,  if  possible,  to  withstand  the  stresses  imposed  upon 
them  if  two-thirds  of  all  the  conductors  in  one  span  are  severed. 
It  must  not  be  overlooked  that  if  the  wires  break  in  one  span  only, 
the  cross-arm,  if  pin  type  insulators  are  used,  will  be  subjected  to 
a  twisting  moment;  and  if  the  break  in  the  wires  is  at  one  end 
only  of  the  cross-arm,  the  whole  tower  is  subjected  to  torsional 
strain. 

The  vertical  or  dead  loads  consist  of  the  weight  of  the  tower 
itself  and  the  wires  of  one  span,  with  possible  increase  in  weight 
due  to  sleet  or  ice.  The  cross-arms  must  be  of  ample  strength  to 
take  all  vertical  loads  including  weight  of  insulators,  with  a  mar- 
gin to  cover  the  extra  weight  of  men  working  on  the  tower. 
The  approximate  weight  of  insulators  is  given  in  the  following 
table: 


336  ELECTRIC  POWER  TRANSMISSION 


Working  line  voltage  Weight  of  insulator,  Ib. 


22,000 
44,000 

•pin  type... 

..:  (j* 

66,000 

40 

88,000 

suspension  type 

55 

110,000 

•  '  70 

140,000 

I  90 

If  a  weight  is  attached  to  the  lowest  unit  in  a  string  of  suspen- 
sion insulators  with  the  object  of  limiting  the  angle  of  deflection 
from  the  vertical  when  a  high  wind  is  blowing  across  the  line, 
this  must  be  taken  into  account  when  determining  the  loads  to 
be  resisted  by  the  supporting  structure. 

Particulars  regarding  wind  pressures  were  discussed  in  Article 
131  of  the  preceding  Chapter.  The  wind  velocity  rarely 
exceeds  80  miles  per  hour  either  on  the  American  continent  or  in 
England.  Tornadoes  and  cyclones  are  not  considered,  because 
attempts  to  design  overhead  lines  strong  enough  to  withstand 
them,  would  hardly  be  justified.  In  regions  where  sleet  de- 
posits are  to  be  expected  it  appears  to  the  writer  unreasonable  to 
base  calculations  on  a  heavier  loading  than  that  described  as 
Class  B  on  p.  271  of  Article  131.  That  is  to  say,  an  ice  coating 
0.5  in.  thick  is  very  rarely  exceeded,  and  when  the  conditions 
are  such  as  to  permit  this  formation,  a  wind  velocity  exceeding 
60  miles  per  hour  (corresponding  to  8  Ib.  per  foot  pressure)  is 
not  likely  to  occur. 

In  regions  where  strong  winds  may  be  expected,  but  where 
sleet  deposits  do  not  occur,  a  maximum  wind  velocity  of  76 
miles  per  hour  seems  a  reasonable  assumption.  This  corresponds 
to  a  pressure  of  21  Ib.  per  sq.  foot  of  flat  surfaces  on  towers, 
and  14  Ib.  per  sq.  foot  of  projected  surface  of  wires  and  cylin- 
drical poles.  The  total  transverse  load  is  dependent  upon 
the  length  of  span,  which  must  be  determined  with  due  regard 
to  economic  considerations. 

163.  Design  of  Steel  Towers. — Although  details  of  design  and 
the  proportioning  of  parts  are  matters  best  left  to  the  manufac- 
turer, the  general  type  of  supporting  structure  to  be  used  under 
given  conditions  should  receive  careful  attention.  The  most 
economical  design  of  tower  to  withstand  the  probable  loads 


TRANSMISSION-LINE  SUPPORTS  337 

that  it  will  be  subject  to,  and  to  satisfy  local  conditions,  includ- 
ing such  considerations  as  transport  and  erection  facilities,  is  a 
problem  deserving  close  attention  on  the  part  of  the  engineer 
responsible  for  the  design  of  the  transmission  line.  A  study  of 
the  probable  loads  to  be  resisted  under  the  worst  weather  con- 
ditions will  enable  the  designing  engineer  to  specify  certain  test 
loads  which  will  ensure  that  the  finished  structure  will  be  strong 
enough  to  fulfil  the  practical  requirements.  The  proper  value  of 
these  test  loads  and  their  distribution  or  point  of  application 
should  be  determined  only  after  mature  consideration.  The 
cost  of  a  tower — apart  from  the  height,  which  is  a  function  of 
the  length  of  span — is  determined  largely  by  the  specifications 
of  test  loads.  A  specification  calling  for  tests  that  are  unneces- 
sarily severe,  is  just  as  true  an  indication  of  incompetence  on  the 
part  of  the  designing  engineer  as  a  specification  giving  test 
conditions  that  will  result  in  a  tower  too  weak  for  the  actual 
requirements. 

The  calculation  of  stresses  in  the  various  members  of  so  simple 
a  structure  as  a  transmission  line  tower  is  not  a  difficult  matter, 
especially  if  graphical  or  semi-graphical  methods  are  adopted. 
If  the  designing  engineer  will  make  sketches  of  two  or  three  al- 
ternative designs  likely  to  fulfil  the  required  conditions,  he  should 
be  able  quickly  to  calculate  the  approximate  value  of  the  stresses 
in  the  principal  members,  and  so  obtain  a  rough  idea  of  the  rela- 
tive weights  and  costs  of  alternative  designs.  The  danger  of 
leaving  the  problem  entirely  in  the  hands  of  the  manufacturer  is 
that  the  latter  is  always  tempted  to  put  forward  a  design  of  which 
he  has  perhaps  made  a  specialty,  and  which  may  have  given 
entire  satisfaction  in  practice  without  necessarily  being  the  best 
type  of  structure  for  the  purpose,  or  being  entirely  suitable  for 
use  under  different  conditions. 

164.  Stresses  in  Compression  Members  of  Tower  Structures. 
— The  failure  of  steel  towers  under  excessive  loads  is  almost 
invariably  due  to  the  buckling  of  the  main  leg  angles  in  compres- 
sion. The  designer  should  therefore  pay  special  attention  to  the 
proportioning  of  compression  members  in  the  structure.  With- 
out going  into  a  discussion  of  the  many  empirical  formulas  used 
for  determining  the  loads  that  struts  or  columns  can  withstand, 
it  may  be  said  that,  for  tower  designs,  the  "straight  line"  for- 
mula, as  suggested  by  Burr,  is  quite  satisfactory  provided  the 
ratio  I  -T-  r  lies  between  40  and  200;  this  last  figure  corresponds 
22 


338  ELECTRIC  POWER  TRANSMISSION 

to  a  length  of  compression  member  not  exceeding  about  twenty 
times  the  width  of  flange.     This  formula  is 

Scomp.  =  K  -  k(l/r)  (156) 

where  K  and  k  are  constants, 

I  is  the  length,  in  inches,  of  unsupported  portion  of  com- 
pression member, 
r  is  the  least  radius  of  gyration,  in  inches, 


ment  of  inertia 
area  of  section 

£COmp.  is  the  unit  stress  (Ib.  per  sq.  in.)  in  the  column. 

The  ultimate  stress  which  will  cause  compression  members  of 
steel  towers  to  collapse  is  approximately  expressed  by  the 
formula, 

Scomp.  =  35,000  -  120  l-  (157) 

Assuming  a  factor  of  safety  of  2^,  we  may  write: 

Safe  working  Scomp.  =  14,000  -  48  l-  (158) 

which  may  be  used  in  the  design  of  steel  towers.  A  similar 
formula,  which  is  in  common  use  for  calculating  safe  loads  in 
compression  members  of  steel  structures  is: 

Scomp.  =  16,000  -  70  l-  (159) 

This  formula  (159)  is  a  safer  one  than  (158)  to  use  when  the 
ratio  l/r  is  large;  but  in  any  case  it  is  recommended  that  l/r 
shall  not  exceed  120  for  main  members  and  150  for  lateral  or 
secondary  members.  The  fact  that,  for  a  given  cross-sectional 
area,  the  shape  of  the  section  is  an  important  factor  in  determin- 
ing the  stiffness  and  ultimate  strength  of  the  members  in  com- 
pression, suggests  that,  where  lightness  and  economy  of  material 
are  of  great  importance,  a  section  of  structural  steel  having  a  large 
moment  of  inertia  per  square  inch  of  cross-section  should  be 
chosen.  The  standard  sections  of  rolled  angles  or  tees  are  some- 
times replaced  by  steel  tubes. 

As  an  example  of  the  relative  economy  of  the  tubular  form  and 
other  forms  of  section,  when  used  as  comparatively  long  struts,  a 


TRANSMISSION-LINE  SUPPORTS 


339 


FIG.  127. — Steel  tower  with  members  of  tubular  section. 


340  ELECTRIC  POWER  TRANSMISSION 

steel  tube  7  in.  internal  diameter,  %  in.  thick,  weighing  10  Ib. 
per  foot,  will  be  as  efficient  in  resisting  compression  as  a  steel 
angle  7^  in.  by  7^  in.  by  J-£  in.  thick,  weighing  25  Ib.  per 
foot,  or  as  an  I  beam  8  in.  by  6  in.  by  ^  in.  thick,  weighing 
35  Ib.  per  foot.  So  large  a  tube  would  not  be  required  except 
in  very  high  towers:  a  tube  from  4  to  5  in.  diameter  would  gen- 
erally be  large  enough  for  the  main  members  of  a  transmission 
line  tower  up  to  100  ft.  high.  The  illustration,  Fig.  127,  is  from 
a  drawing  kindly  supplied  by  Messrs.  Stewarts  and  Lloyds, 
Limited,  of  Glasgow,  Scotland;  it  represents  a  tower  146  feet 
high  as  supplied  for  a  power  transmission  line  in  the  south  of 
England. 

165.  Outline  of  Usual  Procedure  for  Calculating  Stresses  in 
Tower  Members. — The  illustration,  Fig.  128,  which  is  reproduced 
by  kind  permission  of  the  Shawinigan  Water  and  Power  Co., 
and  the  Canadian  Bridge  Co.,  Limited,  shows  a  typical  square- 
base  galvanized  steel  tower  as  used  on  the  Three  Rivers  line  of  the 
Shawinigan  Water  and  Power  Co.  of  Montreal.  These  towers 
are  designed  to  carry  six  aluminum  conductors  of  nineteen 
strand  200,000  circular  mil  cable,  each  being  supported  by  seven 
suspension  disks  of  the  Ohio  Brass  Co.'s  standard  type.  In 
addition  to  the  conductors,  there  are  two  ground  wires  of  %  in. 
stranded  Siemens-Martin  steel  cable  attached  to  the  points 
(1)  at  each  end  of  the  upper  cross-arm.  The  line  is  built  for 
100,000  volts. 

The  method  of  procedure  in  calculating  stresses  is  to  make  a 
sketch  showing  the  points  of  application,  and  the  vertical  and 
horizontal  components,  of  the  outer  forces.  Then  indicate  by 
arrows  the  assumed  horizontal  and  vertical  components  of  the 
reactions,  using  the  suffixes  R  and  L  to  indicate  the  direction  or 
assumed  direction  of  the  horizontal  components.  Since  the 
whole  structure  is  in  equilibrium  under  the  influence  of  the 
various  loads  and  reactions,  it  is  merely  necessary  to  see  that  the 
three  following  conditions  are  satisfied  at  any  point  considered: 

(a)  The  sum  of  all  vertical  force  components  =  zero. 

(b)  The  sum  of  all  horizontal  force  components  =  zero. 

(c)  The  sum  of  all  moments  about  any  point  =  zero. 

When  taking  moments  in  any  particular  plane,  all  those  in  a 
clockwise  direction  would  be  considered  positive  and  those  in  a 
counter-clockwise  direction  negative.  All  joints  are  considered  as 
frictionless  pivots,  which  assumption  is,  of  course,  not  strictly 


TRANSMISSION-LINE  SUPPORTS 


341 


correct,  especially  in  the  case  of  riveted  joints.  It  is  usually 
an  easy  matter  to  choose  a  section  through  the  structure  in 
such  a  position  that  the  stresses  in  a  given  bar  can  readily  be 
calculated  by  applying  one  or  more  of  the  three  equations  of 
equilibrium. 


Top  Vie 


Groundwire  connection  at  2  points-] 

Aluminum  conductor  connection  at  6  polnts-2,3,&4 


Section  C-  C 


Towers  are  designed  for  the  following  Loadings 

I  Breast  pull  of  12000*  horizontal,  normal  or  par- 
allel to  line,  applied  equally  at  four  points  7 
and.  10. 

II  2COO* parallel  to  line  at  each  of  two  points  2,3, 
and  4,  on  one  side  of  tower. 

III  3000   parallel  to  line  at  one  point  1. 

IV  1100*  vertically  at  each  of  three  points  1-2-3  and  4 
V    1100*         "          • eight      ••     12-3  and  4 

I    and  V      are  simultaneous . 

Note: 

All  bolts  X  with  sq.  nut  and  std,  *  10  gave. 

B.  &  8,  washer. 

Holes  punched   "/"  for  K"  bolts. 

All  members  galvanized. 


FIG.  128. — Steel  tower  with  members  of  angle  section. 

The  sketch,  Fig.  129,  will  serve  to  illustrate  the  method  usually 
followed  in  calculating  the  stresses  in  the  main  members  of  a 
tower  structure  such  as  the  one  shown  in  Fig.  128.  The  loading 
considered  is  that  corresponding  to  the  condition  of  test  loads 
I  and  V  applied  simultaneously. 


342 


ELECTRIC  POWER  TRANSMISSION 


The  point  at  which  the  horizontal  breast  pull  of  12,000  Ib.  is 
applied  corresponds  approximately  to  the  point  65  ft.  above 
ground  level  where  the  corner  legs  would  meet  if  produced  beyond 
the  points  (13).     The  weight  of  the  tower  (which  it  is  supposed 
has  not  yet  been  designed  in  detail)  is  taken  at  4000  Ib.,  and  this, 
together  with  the  test  load  V,  gives  a  resultant  vertical  loading  of 
12,800  Ib.  applied  somewhere  on  the  center  line  of  the  tower. 
Consider  a  section  such  as  XY  which  cuts  only  three  members, 
namely,  the  leg  A  at  ground  level, 
the  leg  B  just  above  joint  0',  and 
the  diagonal  brace  C. 

Select  a  point  0  where  the  mem- 
bers A  and  C  meet,  and  consider 
the  moments,  in  the  plane  of  the 
paper,  which  are  produced  about 
this  point  by  the  external  forces 
and  the  reactions  in  the  members 
severed  by  the  imaginary  section 
XY.  It  is  obvious  that  the  stresses 
in  A  and  in  C  have  no  effect  on  the 
tendency  of  the  part  of  the  struc- 
ture above  the  section  line  to  rotate 
on  the  point  0,  and  the  whole  of 
the  externally  applied  turning 
moment  must  be  resisted  by  the 
stress  in  the  member  B.  Therefore 

FK,  ik-Sketch  for  calculation     (12>80°  X  5-75)  +  (12,000  X 

of  stresses  in  tower  members.  47.5)   —   (x  X  11.5)   =  0 

from  which  it  is  found  that  x  =  56,000  Ib. 

Since  there  are  two  members  B  taking  the  whole  crushing  stress, 
the  total  load  tending  to  crush  the  one  member  B  is  28,000  Ib. 
The  length  of  the  unsupported  portion  of  this  member  is  5.5  ft. 
or  66  in.  The  cross-section  of  4  in.  X  4  in.  X  J4  in.  angle  is  1.93 
sq.  in.;  and  the  least  radius  of  gyration,  r  =  0.79.  The  test  load 
should  not  strain  the  tower  beyond  the  elastic  limit.  Using  the 
formula  (157),  the  ultimate  stress  is, 


,.  35,000  -  120  (^ 
:  25,000 


TRANSMISSION-LINE  SUPPORTS  343 

This  corner  member  is  therefore  capable  of  supporting,  just  before 
collapse,  a  compressive  load  of  1.93  X  25,000  =  48,300  Ib.  It 
should  be  of  ample  strength  to  resist  the  test  load  of  28,000  Ib. 
without  permanent  deformation. 

Turning  now  to  the  uplifting  force  acting  in  the  member  A  and 
tending  to  pull  up  the  foundation,  the  center  from  which  the 
moments  are  calculated  is  shifted  to  the  point  0'  where  the  mem- 
bers C  and  B  meet.  The  equation  of  moments  is  now, 

(12,000  X  65)  -  (12,800  X  8)  -  (x  X  16)  =  0 
whence        x  =  42,300 

and  the  tension  in  one  corner  angle  A  is  21,150  Ib. 

The  above  example  briefly  describes  what  is  known  as  the 
method  of  moments.  It  has  been  assumed  that  the  tower  side 
under  consideration  lies  in  the  same  plane  as  the  external  forces: 
but  the  error  introduced  is  practically  negligible.  It  is  an  easy 
matter,  if  desired,  to  make  the  necessary  correction. 

When  calculating  the  stresses  in  a  diagonal  member  such  as  C  of 
Fig.  129,  the  moments  would  be  taken  about  the  point  0", 
which  is  the  junction  of  the  members  A  and  B\  but  in  that  case 
the  actual  loads  on  cross-arms  and  the  wind  pressure  on  the  side 
of  the  tower  would  have  to  be  taken  into  account  and  sub- 
stituted for  the  concentrated  test  load  of  12,000  Ib.  at  the  point 
0"  which  does  not  produce  any  stress  in  the  brace  C  so  long  as  the 
corner  angle  A  remains  truly  straight  and  exerts  no  lateral  pres- 
sure at  the  point  O.  The  method  of  moments  can  usually  be 
applied  for  all  sections  of  a  tower  structure  if  the  imaginary 
dividing  planes  are  properly  placed.  The  counter  members  or 
ties  that  are  not  in  tension  under  the  conditions  of  loading  con- 
sidered are  usually  assumed  to  be  non-existent,  i.e.,  to  serve  no 
useful  purpose  as  compression  members. 

When  computing  the  stresses  in  the  flexible  "A"-frame  steel 
structures  it  is  assumed  that  the  structure  remains  always 
normal  to  the  line  in  a  vertical  plane;  but  unbalanced  forces 
in  the  conductors  will  actually  deflect  the  frame  from  this  position 
and  so  reduce  its  possible  resistance  to  transverse  loads.  It  is 
practically  impossible  to  calculate  the  strength  of  the  distorted 
frame,  and  although  flexibility  in  the  direction  of  the  line  is 
usually  a  desirable  feature  of  this  type  of  structure,  it  is  very 
important  to  design  the  so-called  flexible  steel  towers  so  that 
they  will  not  be  deflected  unduly  by  such  torsional  loads  as  they 


344  ELECTRIC  POWER  TRANSMISSION 

may  be  subjected  to  at  times  when  strong  winds  are  blowing 
across  the  line. 

166.  Stiffness  of  Steel  Towers.  Deflection  Under  Load.— 
The  deflection  of  the  top  of  a  transmission-line  tower  of  the 
ordinary  light  "windmill"  type  with  wide  square  base,  when 
bolted  to  rigid  foundations  and  subjected  to  a  horizontal  load 
such  as  to  stress  the  material  to  nearly  the  elastic  limit,  might  be 
from  2  to  5  in.  With  regard  to  the  two-legged  or  "flexible"  type 
of  tower,  if  this  is  of  uniform  cross-section,  it  may  be  treated  as  a 
beam  fixed  at  one  end  and  free  at  the  other  end.  If  the  resultant 
pull  can  be  considered  as  a  single  concentrated  load  of  P  Ib.  ap- 
plied in  a  horizontal  direction,  at  a  point  H  inches  above  ground 
level,  the  deflection,  in  inches,  will  be, 

s 

(160) 


where  M  is  the  elastic  modulus  for  steel  (about  29,000,000;  being 
the  ratio  of  the  stress  in  pounds  per  square  inch  to  the  extension 
per  unit  length),  and  /  is  the  moment  of  inertia  of  the  horizontal 
section  of  the  structure. 

167.  Tower  Foundations.  —  The  upward  pull  of  the  tower  legs, 
which  was  found  in  the  above  example  to  amount  to  21,150  Ib., 
has  to  be  resisted  by  the  foundation.  A  factor  of  safety  of  2J£ 
to  3  should  be  allowed.  The  weight  of  concrete  may  be  taken 
at  140  Ib.  per  cubic  foot,  and  of  good  earth  at  100  Ib.,  the  volume 
of  the  earth  to  be  lifted  being  calculated  at  the  angle  of  repose, 
which  may  be  about  30  or  33  degrees  with  the  vertical,  as  in- 
dicated in  Fig.  130.  If  the  footing  of  a  tower  is  in  gravel,  or  a 
mixture  of  sand  and  loam  tightly  packed,  there  is  actually  a  far 
greater  resistance  to  the  pulling  up  of  the  footings  than  that  which 
is  offered  by  the  mere  weight  of  the  footings  with  prism  of  earth 
as  calculated  in  the  usual  way. 

When  concrete  has  to  be  used,  it  is  generally  cheaper  to  rein- 
force it  with  steel  of  an  inverted  T  form,  as  this  makes  a  lighter 
construction  than  a  solid  block  of  concrete,  and  an  equally  good 
hold  is  obtained  owing  to  the  increased  weight  of  the  packed 
earth  which  has  to  be  lifted.  At  the  same  time  it  must  not  be 
forgotten  that  the  digging  of  a  large  hole  5  to  8  ft.  deep  is  con- 
siderably more  costly  than  the  digging  of  a  hole  about  2  ft. 
square,  and  this  extra  cost  in  erection  must  be  taken  account  of 
in  designing  the  footings.  In  marshy  or  loose  soil,  or  where  the 


TRANSMISSION-LINE  SUPPORTS 


345 


right  of  way  is  liable  to  be  flooded,  special  attention  should  be 
paid  to  the  design  of  durable  foundations.  Concrete  footings 
with  or  without  piles,  or  rock-filled  crib  work  may  be  necessary; 
it  is  a  matter  requiring  sound  judgment  and,  preferably,  previous 
experience  on  the  part  of  the  engineer  in  charge  of  construction. 
Crumbling  hillsides  are  best  avoided;  it  is  extremely  difficult  to 
guard  against  damage  by  land  slides  or  even  snow  slides  when 
towers  are  erected  on  the  steep  slopes  of  hills. 

The  use  of  concrete  adds  considerably  to  the  cost  of  founda- 
tions and  it  should  be  avoided  if  possible;  on  the  other  hand,  it  is 


Natural  slope  of  \ 

earth  showing  \ 

cone  shaped  mass  \ 

resisting  upward  pull 
on  tower  footing. 


U_._2r_.-^-Bedding  of  large 

>  flat  stones 

FIG.   130.— Foundation  for  steel  tower  anchor  stub. 

not  easy  to  design  foundations  to  resist  a  given  uplift  without 
an  exact  knowledge  of  the  soil  conditions  at  the  site  of  the  tower. 
For  the  greatest  economy  of  foundation,  it  is  necessary  that  the 
designer  obtain  reliable  information  on  this  point. 

Assuming  an  average  angle  of  slope  of  30  degrees,  as  indicated 
in  Fig.  130,  and  a  weight  of  soil  of  100  Ib.  per  cubic  foot,  the 
depth  of,  foundation  may  be  calculated  as  follows. 

Let  h  =  depth  of  footing  below  ground  level,  in  feet. 
r  =  equivalent  radius  of  footing  area,  in  feet. 
R  =  radius  at  ground  level  of  conical  section  of  earth 

to  be  lifted.     (Feet.) 
6  =  angle  of  natural  slope  of  earth. 

The  volume  of  frustrum  of  cone  to  be  lifted  is, 


V  =     h(r*  +  fl2  +  rR) 


(161) 


346  ELECTRIC  POWER  TRANSMISSION 

or,  if  r  +  h  tan  0  be  put  in  the  place  of  R, 

V  =  I  h(3r2  +  h2  tan2  6  +  3r/i  tan  0)  (162) 

If  e  =  30  degrees,  tan  6  =  0.5774  and  (approximately), 

V  =TT  h(r2  +  0.1U2  +  0.58r/i)  (163) 

If  r  =  1  ft.,  and  h  =  7  ft.;  the  volume  of  earth  to  be  lifted,  by 
formula  (163)  is  then, 

V  =  230,  which  gives, 
W  =  23,000  Ib. 

As  previously  mentioned,  if  the  soil  is  firm,  this  method  of  calcu- 
lation usually  gives  results  well  below  actual  values  of  pull  re- 
quired to  uplift  the  footing.  Under  the  conditions  upon  which 
this  example  has  been  based,  it  is  probable  that  the  footing  would 
not  move  with  a  pull  appreciably  smaller  than  30,000  Ib.;  there 
would  then  be  a  packing  of  the  soil  immediately  above  the  footing, 
and  a  final  pull  of  about  40,000  Ib.  might  be  necessary  to  uproot 
the  stub  and  footing. 

If  the  footings  are  imbedded  in  concrete,  and  separate  ground- 
ing rods  are  not  provided,  it  is  well  to  let  the  iron-work  project 
through  the  bottom  of  the  concrete  block,  to  ensure  that  the 
tower  is  properly  grounded. 

When  concrete  is  not  used,  the  design  of  the  anchors  is  a  matter 
that  should  receive  very  careful  consideration.  If  towers  are  to 
be  subjected  to  load  tests,  these  tests  should,  if  possible,  be 
conducted  on  a  tower  set  on  its  own  anchors,  as  used  in  the  field, 
because  the  strength  is  to  an  appreciable  extent  dependent 
upon  the  method  of  attachment  of  the  tower's  legs  to  the  anchor 
stubs.  One  manufacturer  of  transmission-line  towers1  claims 
that  the  old  style  anchor,  with  the  bottom  diagonal  of  the  tower 
attached  to  the  anchor  stub  above  ground,  does  not  afford  proper 
resistance  to  the  horizontal  force  at  the  ground  level  and  some- 
times leads  to  failure  of  the  tower  legs  in  the  bottom  panel. 
This  manufacturing  company  uses  a  design  of  anchor  in  which 
the  joint  between  the  tower  leg  and  the  bottom  diagonal  is  below 
the  ground  surface. 

Another  point  of  importance  is  the  surface  of  the  footing  in 
contact  with  the  earth  immediately  above  it.  The  weight  of  the 

1  The  American  Bridge  Company. 


TRANSMISSION-LINE  SUPPORTS  347 

cone  of  earth  to  be  lifted  may  be  ample  to  provide  the  desired 
factor  of  safety;  but  movement  of  the  tower  foundations  may 
occur  through  the  packing  of  the  earth  due  to  excessive  unit 
pressure  over  the  upper  surface  of  the  footing,  and  this  movement 
may  be  appreciable  notwithstanding  that  there  may  be  no  dis- 
turbance of  the  ground  surface.  A  surface  of  not  less  than 
1  sq.  ft.  for  every  10,000  Ib.  of  the  vertical  force  which  will  pull 
out  the  anchors  should  be  provided,  unless  the  nature  of  the  soil 
is  such  as  to  justify  a  reduction  of  this  allowance. 

168.  Concluding  Remarks  Regarding  Steel  Tower  Design. — 
Generally  speaking,  there  is  a  tendency  to  economize  in  the  cost 
of  steel  towers  by  using  sections  of  structural  steel  in  which 
stiffness  is  obtained  by  making  the  thickness  of  metal  small  in 
proportion  to  the  other  dimensions  of  the  cross-section.  It  is 
true  that  light  weight  of  parts  and  of  the  complete  tower  are 
important  if  the  advantage  of  lightness  can  be  obtained  without 
sacrifice  of  other  advantages,  the  chief  of  which  is  durability. 
When  a  transmission  line  is  not  intended  to  last  longer  than  15 
or  20  years,  these  light  sections  are  permissible;  but  for  the  more 
important  and  costly  lines,  it  is  well  to  avoid  the  use  of  metal 
thinner  than  ^  in.  for  the  main  members,  or  than  %6  in.  for 
the  secondary  or  bracing  members.  In  the  writer's  opinion 
it  is  not  wise  to  use  4"  by  4"  angles  for  the  corner  legs  less  than 
%6  in.  thick,  although  a  thickness  of  ^  in.  is  not  uncommon 
in  towers  actually  in  use  at  the  present  day.  The  ultimate  life 
of  such  towers  is,  however,  as  yet  unknown.  Towers  made  of 
few  pieces  of  comparatively  heavy  section  steel  will  probably 
Drove  more  durable  than  those  built  of  a  larger  number  of  lighter 
parts.  ( 

If  the  temptation  to  use  very  light  sections  of  structural  steel 
is  avoided,  and  if  towers  are  regularly  inspected  and  painted  when 
and  where  necessary,  their  life  should  be  50  years  or  more. 
Galvanized  towers  are  usually  not  painted;  but  it  is  not  safe  to 
rely  upon  the  thin  coating  of  zinc  to  prevent  corrosion  for  more 
than  a  few  years  at  or  near  the  ground  level.  A  casing  of  con- 
crete extending  about  12  in.  above  ground  level  will  afford 
protection;  or  the  parts  that  are  buried  may  be  painted  instead 
of  being  galvanized,  and  if  the  anchor  stubs  are  made  in  two 
lengths,  the  upper  length  can  at  any  time  be  replaced  without 
interrupting  the  service. 

When  considering  designs  of  towers  for  along  transmission  line, 


348  ELECTRIC  POWER  TRANSMISSION 

it  is  well  to  avoid  if  possible  a  number  of  different  types,  and  where 
it  is  not  necessary  to  increase  the  height,  it  may  sometimes  be 
found  more  economical  to  use  two  standard  towers  close  together 
for  supporting  special  long  spans,  or  for  turning  sharp  corners, 
than  to  design  special  towers  for  the  purpose.  An  angle  not  ex- 
ceeding 7  degrees  can  usually  be  turned  on  a  standard  tower. 
This  angle  may  even  be  as  great  as  10  degrees,  especially  if  the 
length  of  the  approach  spans  is  decreased.  In  fact  by  reducing 
the  length  of  approach  spans,  very  much  sharper  angles  can  be 
turned;  but  it  then  becomes  a  question  whether  or  not  a  special 
structure  might  not  be  the  cheaper  alternative. 

There  is  an  unexplained  prejudice  against  the  guying  of  steel 
towers  where  extra  strength  to  resist  lateral  loads  is  required.  By 
giving  proper  attention  to  the  method  of  guying,  and  inspecting 
the  line  at  regular  intervals,  there  is  no  apparent  reason  why  this 
fairly  obvious  device  to  save  the  extra  cost  of  special  structures 
should  not  prove  entirely  satisfactory.  It  is  true  that,  with  the 
so-called  rigid  design  of  tower,  a  very  small  deflection  at  the  point 
of  attachment  of  the  guy  wire  may  be  sufficient  to  produce 
permanent  deformation  of  the  structure,  and  there  is  a  possibility 
that  the  tower  may  collapse  under  excessive  load  before  the  guy 
wires  have  taken  up  their  proper  share  of  the  abnormal  stresses. 
This  is  especially  likely  to  occur  if  the  towers  are  set  on  concrete 
foundations.  On  the  other  hand  it  is  not  impossible  to  design 
towers  of  the  square  base  type  with  foundations  purposely 
arranged  to  yield  slightly;  and  if  these  structures  are  provided 
with  guys  (say  of  plow  steel  cables)  fixed  very  securely  to  practi- 
cally unyielding  concrete  anchorages,  it  is  probable  that  econo- 
mies might,  in  many  instances,  be  effected.  Guying  of  corner 
poles  or  of  occasional  poles  on  a  straight  run,  when  the  more 
flexible  type  of  "A "-frame  structure  is  used,  is  generally  to  be 
recommended. 

A  brief  specification  for  a  complete  transmission  line  using 
steel  towers  is  given  in  Appendix  III.  This  line  is  generally  simi- 
lar to  the  one  for  which  an  estimate  of  cost  was  given  in 
Chapter  III. 

169.  Determining  Position  of  Supports  on  Uneven  Ground. — 
The  lowest  point  of  the  span  is  not  necessarily  the  point  at  which 
the  wires  come  closest  to  the  ground.  When  there  is  doubt  as 
to  the  proper  location  of  the  supports  in  rough  country,  the 
method  illustrated  in  Fig.  131,  and  described  by  Mr.  J.  S.  Viehe 


TRANSMISSION-LINE  SUPPORTS 


349 


in  the  Electrical  World  of  June  15,  1911,  will  be  found  very  con- 
venient. The  curve  a  is  the  parabola  corresponding  to  the  re- 
quired tension  in  the  particular  wire  to  be  used.  The  ratio  of  the 
scale  of  feet  for  vertical  measurements  to  the  scale  for  horizontal 
measurements  should  be  about  10  to  1.  The  dotted  curves  6 
and  c  are  exactly  similar  to  a,  but  the  vertical  distance  ab  repre- 
sents the  minimum  allowable  clearance  between  conductor  and 
ground,  while  the  vertical  distance  ac  is  the  height  above  ground 
level  of  the  point  of  attachment  of  the  lowest  wires  to  the  stand- 
ard transmission  pole  or  tower.  These  curves  should  be  drawn 
on  transparent  paper:  they  can  then  be  moved  about  over  a 


FIG.  131. — Method  of  locating  position  of  towers  in  rough  country. 

profile  of  the  ground  to  be  spanned,  drawn  to  the  same  scale  as 
the  curves,  until  the  best  location  for  the  supports  is  found.  The 
point  P  where  the  curve  6  touches  the  ground  line  is  seen  to  be 
far  removed  from  the  lowest  point  of  the  parabola,  in  the  example 
illustrated  in  Fig.  131.  A  little  practice  will  make  the  finding 
of  the  points  A  and  B  an  easy  matter,  even  if  the  length  of  span, 
or  distance  between  A  and  B,  must  be  kept  between  close  limits. 

This  method  is  particularly  applicable  to  long-span  lines 
carried  over  rough  country. 

170.  Study  of  Deflections  and  Stresses  in  Flexible  Tower 
Lines. — Consider  a  series  of  poles  as  in  Fig.  132,  the  end  one  being 
rigid  while  all  the  others  are  flexible  and  of  equal  height  and  stiff- 
ness. It  is  assumed  that  all  spans  were  originally  of  equal 
length  I,  and  that  there  were  6  wires  in  each  span,  strung  to  a 


350 


ELECTRIC  POWER  TRANSMISSION 


tension  of  S  pounds  per  square  inch  and  having  a  corresponding 
sag  s.  In  span  No.  1,  terminating  at  the  rigid  support,  some  of 
the  wires  have  been  severed,  leaving  only  a  wires  in  this  span. 
It  is  assumed  also  that  there  is  no  slipping  of  the  wires  in  the  ties 
on  the  pin  type  insulators,  and  no  yielding  of  pole  foundations. 

The  elastic  deflection  of  a  pole  or  tower  considered  as  a  beam 
fixed  at  one  end  and  loaded  at  the  other  is 

PH* 


ZMI 

where  P  is  the  load,  H  the  height,  M  the  elastic  modulus,  and  7 
the  moment  of  inertia  of  the  cross-section. 

In  the  special  case  considered,  the  value  of  P,  which  produces 
the  deflection  5i  of  the  first  flexible  pole,  is 

P  =  A(bS,  -  aSi) 


b  Wires 


_       I  u  u  u 

FIG.   132. — Flexible  pole  line. 

where  A  is  the  cross-section  of  one  conductor  and  Si  and  S%  are 
the  stresses  in  the  conductors  of  spans  No.  1  and  No.  2  respect- 
ively. It  is  assumed  that  all  the  wires  are  attached  to  the  pole 
tops  at  a  point  H  in.  above  ground  level. 

By  putting  K  =5-577'  the  successive  deflections  maybe  written: 


81  =  KA(bS2  -  aSi)  (164) 

62  =  KAb(S»  -  S2)  (165) 

and  the  sum  of  the  deflection  of  a  series  of  flexible  poles  of  the 
same  height  and  stiffness  is 

A  =  KA(bSn  -  oSi)  (166) 

where  n  is  the  number  of  the  last  span.  It  is  usually  safe  to  as- 
sume that  Sn  is  equal  to  the  initial  tension  S  in  the  fourth  or  fifth 
span  from  the  break. 


TRANSMISSION-LINE  SUPPORTS  351 

Fig.  133  shows  the  conductors  in  the  first  span  with  a  sag  s 
under  normal  conditions  with  b  wires  in  the  span,  and  a  smaller 
sag  Si  after  some  of  the  wires  have  been  cut,  leaving  only  a  wires 
in  the  span.  For  simplicity  in  calculating  the  movement  of  the 
point  of  attachment  of  the  wires  on  the  flexible  pole,  instead  of 
considering  the  span  as  increasing  in  length  from  I  to  (Z  +  5), 
the  span  I  may  be  supposed  to  remain  unaltered  while  the  length 
of  the  conductor  is  reduced  by  pulling  it  through  the  tie  of  the 
insulator  (G)  on  the  flexible  pole  until  the  sag  is  reduced  from  s 
to  s\.  The  length  of  wire  pulled  through  in  this  manner  may, 
for  all  practical  purposes,  be  considered  equal  to  the  actual  pole- 
top  deflection,  8.  This  assumption  is  justifiable  since  the  de- 
flection 8  is  always  small  relatively  to  the  span  I. 

The  length  of  the  (parabolic)  arc  with  sag  s  is 


FIG.   133. — Elongation  of  wire  in  span  due  to  deflection  of  pole  top. 

and  with  sag  Si 

The  difference  is 

_       _  8(s2  —  si2) 
~37~ 

to  which  must  be  added  the  elongation  due  to  the  stretch  of 
the  wire  under  increased  tension;  this  is 

Xi  (Si  -  S} 
M  ~ 

or,  with  quite  sufficient  closeness, 

l(Si  —  S) 

~W~ 


352  ELECTRIC  POWER  TRANSMISSION 

Hence  the  deflection  of  the  first  flexible  pole  expressed  in  terms 
of  the  sag  and  tension  of  the  conductors  in  the  first  span  is  : 


(167) 


Returning  again  to  the  arrangement  of  line  depicted  in  Fig.  132, 
we  shall  consider  (1)  the  total  pull  of  all  the  wires  in  span  No.  2 
and  the  effect  of  this  pull  on  the  first  flexible  pole  if  all  the  wires 
are  broken  in  span  No.  1,  and  (2)  the  effect  on  the  first  flexible 
pole  and  the  stresses  in  the  remaining  wires  in  No.  1  span  on  the 
assumption  that  all  the  wires  in  this  span  are  not  broken. 

When  the  particulars  of  the  poles  are  known,  so  that  the  factor 
K  in  the  formulas  for  deflection  can  be  determined,  it  is  desired  to 
calculate  the  stresses  in  poles  and  wires  corresponding  to  the  new 
conditions  of  equilibrium;  or,  if  the  poles  have  yet  to  be  designed, 
the  factor  K  must  be  determined,  in  order  that  the  stiffness  of  the 
poles  shall  satisfy  certain  necessary  or  assumed  conditions,  such 
as  the  maximum  deflection  of  pole  top  which  will  not  stress  the 
remaining  wires  in  span  No.  1  beyond  the  elastic  limit  of  the 
conductor  material.  (A  factor  of  safety  must  be  used  to  allow  of 
momentary  increased  stresses  due  to  probable  mechanical 
surges.) 

171.  Numerical  Example:  Transmission  Line  with  Flexible 
Supports.  —  No  attempt  will  be  made  to  obtain  an  exact  mathe- 
matical solution  of  these  problems,  but  close  approximations  can 
be  obtained  with  sufficient  accuracy  for  practical  purposes,  espe- 
cially when  it  is  considered  that  many  possible  influencing  factors, 
such  as  the  yielding  of  foundations  and  the  slipping  of  wires  in  the 
ties,  cannot  be  taken  into  account  even  in  the  most  complete 
mathematical  treatment  of  the  subject. 

It  is  assumed  that  the  poles  are  equidistant  and  in  a  straight 
line,  and  that  the  first  support  is  rigid,  all  as  indicated  in  Fig. 
132.  Four  separate  limiting  conditions  will  be  considered  : 

(A)  All  wires  are  severed  in  the  first  span,  and  the  pole  between 
spans  2  and  3  is  considered  to  be  rigid. 

(B)  All  wires  are  severed  in  the  first  span,  but  the  pole  between 
spans  2  and  3,  and  all  subsequent  poles,  are  considered  to  offer 
no  resistance  to  deflection  in  the  direction  of  the  line. 

(C)  There  are  a  wires  remaining  in  span  No.  1,  and  6  wires  in 
all  other  spans.     The  pole  between  spans  2  and  3  is  considered  to 
be  rigid. 


TRANSMISSION-LINE  SUPPORTS  353 

(D)  There  are  a  wires  in  the  first  span,  but  the  pole  between 
spans  2  and  '3,  and  all  subsequent  poles,  are  considered  to  be 
infinitely  flexible. 

The  transmission  line  will  be  supposed  to  have  the  following 
characteristics: 

Six  No.  2-0  aluminum  conductors. 

Cross-section  of  conductor,  A  =  0.1046  sq.  in. 

Length  of  span,  I  =  400  ft. 

Normal  sag  =  9.76  ft.,  which  corresponds  to 

Stress  S  =  2400  Ib.  per  square  inch. 

It  is  assumed  that  there  is  no  grounded  guard  wire  above  the 
conductors,  and  that  the  average  height  of  the  point  of  attach- 
ment of  the  wires  above  ground  level  is  H  =  45  ft. 

The  modulus  of  elasticity  of  aluminum  cables  for  the  purpose 
of  these  calculations  is  assumed  to  be  M  =  7,500,000.  The  flexi- 
ble towers  are  in  the  form  of  braced  "A"-frames,  each  vertical  limb 
consisting  of  one  7-in.  steel  channel  of  light  section  (9%  Ib.  per 
foot)  .  The  moment  of  inertia  of  the  section  of  such  a  channel  is 
21.1,  and  since  there  are  two  channels,  the  value  of  7  is  21.1  X  2 

42  2 
=   42.2   and   the  section  modulus  Z=  ~r  =  (say)   12.     The 

elastic  modulus  for  steel  is  M  =  29  X  106.  The  factor  K  for  use 
in  pole  deflection  formulas  as  previously  given  is  therefore 

_  (45  X  12)3  _ 

~  3  X  29  X  10«  X  42.2  ~ 

The  maximum  deflection  of  this  particular  structure  before 
permanent  deformation  would  take  place  will  occur  when  the 
difference  of  pull  due  to  the  wires  is  such  as  to  stress  the  metal 
to  (say)  30,000  Ib.  per  square  inch.  The  resisting  moment  is 
S  X  Z  =  30,000  X  12  and  the  resultant  pull  at  the  pole  top  will 


The  maximum  allowable  deflection  is  therefore, 

5=  K  X  667 
=  0.0428  X  667 
=  28.5  inches. 

Case  (A).     All  wires  broken  in  span  No.  1;  second  pole  beyond 
break  considered  rigid. 


354 


ELECTRIC  POWER  TRANSMISSION 


Since  all  the  wires  are  severed  in  span  No.  1  (a  =  0)  it  is  not 
possible  to  make  use  of  formula  (167),  but  a  similar  formula  can 
be  used  which  expresses  the  deflection  in  terms  of  the  constants 
for  span  No.  2.  This  formula  is 


.8 


(168) 


By  calculating  61  for  various  arbitrary  values  of  £2  smaller  than 
S,  curve  No.  1  of  Fig.  134  can  readily  be  drawn.  This  gives  the 
relation  between  the  stress  Sz  in  the  wires  of  the  second  span  and 
the  pole-top  deflection  5i  on  the  assumption  that  the  second 
pole  beyond  the  break  is  rigid.  On  the  same  diagram  draw  the 


200  1400  1600  1800  2000  2200 

Stress  in  wires  of  span  No.2  =  Sz  Jbs.  .per.sq.  inch 

FIG.   134. — Graphic  solution  of  Problems  (A)  and  (B). 


2400 


straight  line  marked  curve  No.  2,  which  gives  the  relation  between 
pole  deflection  and  the  stress  $2,  as  given  by  formula  (164)  when 
the  tension  S:  in  wires  of  the  first  span  is  equal  to  zero.  The 
point  of  crossing  of  curves  No.  1  and  No.  2  evidently  indicates 
the  deflection  corresponding  to  the  condition  of  equilibrium. 
This  deflection  is  Si  =  29.5  in.  and  stress  S2  =  1100. 

It  will  be  noted  that  in  this  particular  example  the  deflection 
is  about  the  same  as  the  maximum  permissible  deflection  (28.5) 
previously  calculated ;  but  even  if  allowance  be  made  for  shocks 
and  mechanical  surges,  it  is  probable  that  the  pole  would  not 
suffer  serious  injury,  because  some  of  the  wires  would  be  liable  to 


TRANSMISSION-LINE  SUPPORTS  355 

slip  in  the  ties  and  so  relieve  the  tension.  If  wind  pressures  acting 
on  snow  or  ice  deposits  are  added  to  the  stresses  due  to  weight  of 
conductor  material  only,  the  strain  will  be  greater,  but  on  the 
other  hand,  much  sleet  deposit  is  liable  to  be  shaken  off  the  wires 
in  the  event  of  a  sudden  severing  of  all  the  wires  in  the  first 
span. 

The  above  results  are,  however,  based  on  the  assumption  that 
the  second  pole  beyond  the  break  is  rigid,  which  may  not  be  in 
accordance  with  practical  conditions. 

Case  (B).  Conditions  as  above;  but  the  second  and  subse- 
quent poles  beyond  the  break  are  supposed  to  be  infinitely  flexible 
(K  =  »). 

In  this  case  the  tension  Sz  will  not  depend  upon  the  deflection 
of  the  first  flexible  pole;  it  will  be  equal  to  the  original  tension  S 
=  2400  for  all  values  of  the  deflection  Si.  The  deflection  ob- 
tained when  Sz  =  2400  is  of  course  readily  calculated  by  means 
of  formula  (164),  or  it  can  be  read  off  Fig.  134,  since  it  is  the 
deflection  indicated  at  the  point  where  curve  No.  2  meets  the 
vertical  ordinate  for  Sz  =  2400.  This  value  of  §1  is  64.5  in., 
which  would  lead  to  permanent  deformation  of  the  flexible 
structure.  The  actual  deflection  of  the  first  pole  in  a  series  of 
flexible  poles  of  equal  stiffness  would  lie  somewhere  between 
these  limiting  values  of  29.5  in.  and  64.5  in.  if  the  law  of  elasticity 
may  be  considered  to  apply  in  the  case  of  the  higher  deflections. 
As  a  general  rule  the  breaking  of  all  wires  hi  one  span  will  lead 
to  the  wrecking  or  permanent  deflection  or  uprooting  of  the 
first  pole,  which  cannot  be  at  the  same  time  flexible  enough 
greatly  to  reduce  the  combined  pull  of  all  wires  in  span  No.  2, 
and  yet  strong  enough  to  resist  the  ultimate  combined  pull  of 
these  wires.  There  would  be  an  exception  in  the  case  of  short 
spans  with  tall  flexible  poles;  and  in  any  case  it  is  probable  that 
only  the  first  pole  would  be  damaged  or  moved  in  its  foundations. 

It  is  rare  that  all  the  wires  in  one  span  are  broken  simulta- 
neously unless  the  design  of  the  line  is  such  that  the  severing  of 
one  or  more  wires  leads  necessarily  to  the  rupture  of  the  remaining 
wires  owing  to  the  excessive  stresses  imposed  on  them.  The  cal- 
culation of  stresses  and  deflections  when  a  certain  number  of 
wires  remain  in  the  faulty  span  is  more  difficult  than  in  the  cases 
already  considered,  but  the  solution  is  of  greater  practical  value. 

Case  (C).  There  are  a  wires  in  the  faulty  span  and  6  wires  in 
the  sound  spans.  The  second  pole  beyond  break  is  considered 


356  ELECTRIC  POWER  TRANSMISSION 

rigid.  (For  the  purpose  of  working  out  numerical  examples 
it  will  be  assumed  that  only  one  wire  remains  in  faulty  span; 
thus  a  =  1  and  b  =  6.) 

Instead  of  only  two  equations,  there  are  now  three  equations 
to  be  satisfied  simultaneously;  these  are: 

(a)  Formula  (164): 

81  =  KA  (bS2  -  aSi) 

=  0.0269S2  -  0.00448Si 

(b)  Formula  (167),  giving  deflection  in  terms  of  elongation  of 
remaining  wires  in  span  No.  1: 


(c)  Formula  (168),  giving  deflection  in  terms  of  the  shortening 
of  the  wires  in  span  No.  2.  (This  relation  is  given  by  curve  No.  1 
already  plotted  in  Fig.  134.) 

It  should  be  mentioned  in  connection  with  formulas  (167) 
and  (168)  that,  by  assuming  a  constant  length  of  span,  the  sag 
s  is  always  inversely  proportional  to  the  stress  S.  The  assump- 
tion of  a  constant  length  of  span  for  the  purpose  of  simplifying 
the  relation  between  sag  and  tension  introduces  no  appreciable 
error  in  practical  calculations.  In  the  particular  example  from 
which  the  curves  are  plotted,  and  the  numerical  results  obtained, 

23,420 

the  relation  is  s  =  —  ~  —  • 
o 

Proceed,  now,  to  plot  curve  No.  3  in  Fig.  135  from  formula 
(167)  by  assuming  various  arbitrary  values  of  Si  from  the  lowest 
possible  limit  of  Si  =  S  =  2400  up  to  the  elastic  limit  of  about 
13,000.  For  a  reason  to  be  made  clear  hereafter  this  curve  should 
be  drawn  on  transparent  paper;  the  horizontal  scale  used  for  the 
values  of  Si  may  be  arbitrarily  chosen,  but  the  scale  of  ordinates 
giving  the  deflections  5i  must  be  exactly  the  same  as  used  for 
Fig.  134.  On  the  same  diagram  (Fig.  135)  draw  also  the  straight 
line  marked  curve  No.  4,  giving  the  relation  between  Si  and  the 
quantity  KAaSi.  This  latter  quantity  when  subtracted  from 
the  quantity  KAbS2  will  give  the  pole  deflection  to  fulfil  the 
condition  of  formula  (164).  The  reason  for  drawing  the  curves  of 
Fig.  4  on  transparent  paper  will  now  be  clear. 

The  transparent  paper  with  the  curves  of  Fig.  135  is  placed 
over  Fig.  134,  with  the  horizontal  datum  lines  of  zero  deflection 


TRANSMISSION-LINE  SUPPORTS 


357 


coinciding  as  shown  in  Fig.  136.  The  point  of  intersection  of 
curves  No.  1  and  No.  3  will  give  the  corresponding  values  of  the 
stresses  Si  and  Szl  but  with  a  pole  having  definite  elastic  prop- 


I- 


£10 
o 


10000  12000 

Stress  in  wires  of  span  No.l  =  S\  Ibs.  per  uq.  inch 


FIG.   135.— Curves  to  be  drawn  on  tracing  paper  for  solution  of  Problems  (C)  and 

(D). 


Lower  Sheet  with  Curves  Nos.l  and  2 


Transparent  Paper  with  Curves  Nos.3  and  4 
FIG.  136.—  Graphic  solution  of  Problems  (C)  and  (D). 

1  There  is  a  definite  value  of  Si  for  any  given  value  of  £2  independent  of 
all  considerations  of  pole  stiffness  and  size  of  wire  and  number  of  wires  in 
adjoining  spans.  This  is  the  relation  which  will  satisfy  formulas  (167)  and 
(168)  simultaneously;  it  is  expressed  by  the  equation 


(2s*  -  8l»  - 


358  ELECTRIC  POWER  TRANSMISSION 

erties  there  is  only  one  value  of  the  deflection  which  will  satisfy 
the  three  conditions  previously  referred  to.  The  deflection  as  a 
function  of  the  pole  stiffness  is  the  distance  EF  (Fig.  136), 
being  the  difference  between  the  corresponding  ordinates  of 
curves  No.  2  and  No.  4.  By  moving  the  tracing  paper  with  the 
curves  No.  3  and  No.  4  over  the  other  curves  until  the  distances 
HG  and  FE  on  the  same  vertical  ordinate  are  equal, -the  deflection 
corresponding  to  the  condition  of  equilibrium  is  readily  obtained. 
If  preferred,  the  curve  OPRE,  representing  the  sum  of  the 
quantities  of  curves  No.  3  and  No.  4,  may  be  drawn  on  the 
tracing  paper  instead  of  the  curve  4,  and  when  the  point  of 
intersection  (E)  of  this  new  curve  with  curve  No.  2  on  the  lower 
sheet  lies  on  the  same  vertical  ordinate  as  the  junction  ((7)  of 
the  curves  No.  1  and  No.  3,  the  distance  HG  will  be  the  required 
deflection. 

The  solution  of  the  numerical  example  worked  out  in  this 
manner  is 

81  =  10.2  in. 

51  =  7400  Ib.  per  square  inch. 

52  =  1500  Ib.  per  square  inch. 

Case  (D).  Same  conditions,  with  the  exception  that  the 
second  pole  beyond  the  break,  instead  of  being  rigid,  is  assumed 
to  be  infinitely  flexible.  This  assumption  is  made  also  in  the 
case  of  all  subsequent  poles.  This  means  that  £2  =  S  =  2400 
whatever  may  be  the  amount  of  deflection  of  the  first  flexible 
pole,  and  the  problem  can  be  solved  graphically  as  indicated 
above,  the  only  difference  being  that  curve  No.  1  giving  the 
relation  between  81  and  $2  when  the  second  pole  beyond  the 
break  is  rigid  must  be  replaced  by  the  vertical  line  SW  (Fig.  136), 
being  the  ordinate  corresponding  to  a  tension  S2  =  2400. 

The  numerical  solution  in  this  case  is: 

di  =  13.2  in. 

Si  =  ll,4501b.  persq.  in. 

It  is  interesting  to  note  that  there  is  little  difference  between  the 
deflections  for  the  two  extreme  cases  (C)  and  (Z>);  the  average 
value  for  5i  is  11.7  in.,  corresponding  to  a  stress  Si  =  9400  in  the 
remaining  wire  of  the  faulty  span.  This  is  well  below  the  elastic 
limit,  and  it  is  probable  that  this  wire  would  not  break  even  if  the 
five  other  wires  were  severed.  The  figures  chosen  for  illustrating 


TRANSMISSION-LINE  SUPPORTS  359 

the  calculations -relate  to  a  practical  transmission  line,  and  it  will 
be  seen  that  the  stresses  and  deflections  corresponding  to  the 
state  of  equilibrium  after  the  severing  of  one  or  more  wires  in  one 
span  can,  with  the  help  of  simple  diagrams,  be  predetermined 
within  reasonably  narrow  limits. 

172.  Erection  of  Steel  Tower  Transmission  Lines. — This  book 
is  not  intended  to  give  practical  advice  to  construction  engineers 
or  the  men  actually  engaged  in  the  work  of  erecting  poles  or  towers 
and  stringing  wires.  A  competent  construction  engineer  with 
experience  in  handling  men  and  materials  in  the  field,  should  be 
given  a  free  hand  in  planning  and  executing  the  work  of  erecting 
a  power  transmission  line;  and  such  a  man  will  not  derive  much 
assistance  from  books.  On  the  other  hand,  there  are  some  excel- 
lent books  available  dealing  with  the  more  practical  side  of  trans- 
mission line  engineering.  These  include  the  various  electrical 
engineering  handbooks.  The  reader  desiring  information  on  the 
methods  ordinarily  adopted  in  carrying  out  the  details  of  con- 
struction, is  referred  to  these  other  sources  of  information;  also 
to  the  papers  and  articles  which  appear  from  time  to  time  in  the 
Journals  of  the  engineering  societies  and  in  the  technical  press. 
Appendices  II  and  III  which  follow  this  Chapter  also  contain 
items  of  some  practical  interest  connected  with  the  setting  out 
and  erection  of  wood  pole  and  steel  tower  lines. 

The  principal  reason  for  referring  to  these  matters  in  this 
place  is  to  emphasize  the  importance  of  devoting  much  time  and 
thought  to  the  various  details  of  overhead  line  construction 
before  the  work  is  actually  started.  The  proper  setting  out  of  the 
line  is  among  the  most  important  matters  connected  with  over- 
head construction.  If  a  line  is  not  carefully  surveyed  and 
planned  in  every  detail,  it  will  often  be  impossible  to  get  good 
and  reliable  service  from  it.  This  does  not  mean  that  the  commer- 
cial aspect  of  the  undertaking  is  not  of  prune  importance;  on  the 
contrary,  it  is  the  only  aspect  from  which  an  engineering  under- 
taking of  the  kind  under  consideration  should  be  viewed.  But 
this  is  not  equivalent  to  saying  that  a  small  first  cost  is  always 
desirable,  or  that  even  a  short  low-voltage  transmission  line  can 
be  constructed  and  operated  economically  by  persons  without 
engineering  skill  and  experience.  It  is  an  easy  matter  to  find 
examples  of  lines  that  have  cost  too  much;  but  it  is  not  impossible 
to  find  the  transmission  line  that  has  cost  too  little — in  the  first 
instance. 


360  ELECTRIC  POWER  TRANSMISSION 

Generally  speaking,  the  writer  believes  that  not  enough  atten- 
tion is  paid  to  preliminary  investigations  and  estimates  of  power 
transmission  lines.  The  construction  of  comparatively  short 
lines  for  moderate  voltages  appears  to  be,  and  indeed  is,  a  fairly 
simple  piece  of  work;  yet — in  respect  to  economy  and  service — 
such  lines  may  be  a  source  of  endless  trouble  if  they  have  been 
planned  and  constructed  without  regard  to  the  fundamental 
principles  of  engineering. 


APPENDIX  I 

INDUCTANCE    OF    TRANSMISSION    LINES    WITH    ANY 
ARRANGEMENT  OF  PARALLEL  CONDUCTORS1 

The  manner  in  which  the  inductance  and  the  induced  e.m.f. 
can  be  calculated  when  the  conductors  of  a  three-phase  system 
occupy  the  vertices  of  an  equilateral  triangle,  was  explained 
in  Chapter  II;  and  it  was  also  stated  that  a  departure  from  the 
symmetrical  arrangement  of  conductors  does  not  modify  the 
calculated  results  to  a  great  extent.  It  will  be  interesting  to 
study  the  problem  in  its  broader  aspect,  with  a  view  to  ascer- 
taining what  is  the  nature  and  magnitude  of  the  modifying  fac- 
tors. It  is  proposed  to  indicate  a  simple  method  of  calculating 
the  total  induced  e.m.f.  in  any  conductor  of  an  electric-energy 
transmission  system,  whatever  may  be  the  actual  arrangement 
or  relative  positions  of  the  conductors.  It  is  assumed  in  all 
cases  that  the  conductors  are  of  circular  section  and  that  they 
remain  parallel  with  each  other  throughout  the  whole  distance 
of  transmission. 

Calculation  of  Total  Resultant  Flux  Surrounding  One  Con- 
ductor When  There  Are  Several  Return  Conductors. — In  Fig.  1 
the  total  outgoing  current  /  is  supposed  to  flow  along  one  con- 
ductor, while  the  total  return  current  is  divided  between  a  number 
of  conductors,  the  condition  being  that 

7    =     -(A   +   /2   +   /3 +/n) 

Let  di,  dz,  d3,  etc.,  represent  the  distances  between  centers  of 
the  corresponding  conductors  carrying  the  return  currents  and 
the  conductor  carrying  the  outgoing  current,  and  note  that  the 
total  flux  surrounding  the  latter  conductor  may  be  considered 
as  the  algebraic  sum  of  several  separate  fluxes,  namely,  the  flux 
due  to  a  current  I\  returning  at  a  distance  d\]  the  flux  due  to  a 
current  72  returning  at  a  distance  dz,  and  so  on,  for  any  number  of 
components  of  the  total  current  7.  All  these  separate  compo- 
nents of  the  total  flux  can  readily  be  calculated  by  means  of 
formula  (25)  of  Article  43,  Chapter  IV,  and  the  expression  for  the 
total  flux  surrounding  a  conductor  in  which  the  current  7  returns 

1  This  Appendix  is  a  reprint,  with  slight  changes  and  omissions,  of  articles 
which  were  first  published  in  the  Electrical  World  of  May  23,  1908  and  Sept. 
15,  1910. 

361 


362  ELECTRIC  POWER  TRANSMISSION 

along  a  number  of  separate  conductors,  as  indicated  in  Fig.  1, 
becomes  : 

2Z  T          r      i          di  T     ,          dz  r     ,          dn~\         ,.. 

*  -  •jjjl"-    *     gt  7       2    ge7  .....  ~   n    ge  7J     (  } 


where  r  stands  for  the  radius  of  cross-section  of  the  conductor 
carrying  what  will  be  thought  of  as  the  outgoing  current  /. 

In  the  case  of  energy  transmission  by  polyphase  currents,  with 
any  number  of  conductors,  the  algebraic  sum  of  the  currents  in 
the  conductors  must,  at  any  given  instant,  be  equal  to  zero. 
Any  one  conductor  may  be  looked  upon  as  carrying  the  outgoing 
current,  while  the  remaining  conductors  together  carry  the  return 
current.  Formula  (1)  can,  therefore,  be  used  for  calculating 
the  effective  flux  of  induction  surrounding  any  one  conductor  in 
a  polyphase  transmission,  whatever 
may  be  the  arrangement  of  the  con- 
ductors. The  phase  relations  of  the 
various  component  fluxes  must,  how- 
ever, be  taken  into  account,  and  for 

FIG.     1.  —  Section    through         ,.  ,  i-i        ITJ-  r 

four  parallel  conductors.  this  reason  the  graphical  addition  ot 
vector  quantities  with  the  help  of  a 

diagram  will  be  found  most  convenient.  Instead  of  drawing  the 
vectors  representing  magnetic  flux  components  —  in  phase  with 
the  current  vectors  —  the  component  vectors  of  the  resulting 
e.m.f.  of  self-induction  may  be  drawn  —  in  this  case  90  time-de- 
grees behind  the  corresponding  current  vectors. 

Calculation  of  E.M.F.  of  Self-  and  Mutual  Induction.—  In 
order  to  calculate  the  induced  e.m.f.  it  will  be  advisable  first  to 
put  equation  (1)  in  a  more  practical  form.  The  symbols  /i,  1  2, 
etc.,  in  equation  (1),  when  the  latter  is  to  be  used  for  calculating 
the  maximum  value  of  the  induction  due  to  an  alternating  cur- 
rent, must  be  considered  as  representing  the  maximum  value  of 
the  current  wave;  but  it  will  be  more  convenient  to  assume 
sinusoidal  currents,  and  then  let  these  symbols  stand  for  the 
virtual  (or  r.m.s.)  value  of  the  currents. 

The  procedure  is  now  as  indicated  in  Article  45  of  Chapter  IV, 
leading  up  to  formula  (28)  which  may  be  written: 

Reactive  volts  per  mile  |         _  ,         d  .   ,T  /m 

,.    •     ,  =  al  log™  -  +  bl  (2) 

of  single  conductor     j  r 

where  a  =  0.00466/ 
and  b  =  0.000506/ 


INDUCTANCE  OF  TRANSMISSION  LINES       363 

The  item  bl  is  ihe  reactive  voltage  component  due  to  the  flux 
set  up  inside  the  material  of  the  conductor  by  the  current  /. 
The  flux  producing  this  increased  reactive  e.m.f.  is  not  included 
in  the  flux  as  calculated  by  formula  (1).  It  may  generally  be 
neglected  in  calculations  of  high  voltage  overhead  transmission 
lines. 

The  final  expression  for  the  reactive  voltage  drop  per  mile  of 
conductor  when  there  are  several  parallel  return  conductors  is: 

E  =  a  [-  I,  log  £  -  72  logy2  .  .  .  .  -  In  logy"]  +  67     (3) 

Numerical  Example.  Three-phase  Transmission. — Consider 
the  special  case,  which  not  infrequently  arises  in  practice,  of  the 
conductors  of  a  three-phase  transmission  being  arranged  as  indi- 
cated in  Fig.  2 — that  is,  with  the  centers  of  the  three  conductors 
lying  in  the  same  plane,  the  minimum  distance,  d,  between  any 
two  of  the  wires  being  approximately  equal  to  the  side  of  the 
equilateral  triangle  which  would  have 

* ^ _3         been  adopted  had  the  triangular  ar- 

U d »H d *^2r    rangement  been  decided  upon. 

FIG.  2.-Three  conductors  in        In  a  three-phase  transmission  system 

one  plane.  the  current  flowing  out  through  any 

one  wire  may,  as  previously  mentioned, 

be  considered  as  returning  along  the  two  remaining  wires,  and  when 
the  three  conductors  occupy  the  vertices  of  an  equilateral  triangle 
the  whole  of  the  return  current  is  at  a  distance  d  from  the  out- 
going current.  This  condition  also  applies  to  the  middle  con- 
ductor (No.  2)  in  the  arrangement  shown  in  Fig.  2;  but  it  does  not 
apply  to  either  of  the  outside  conductors,  Nos.  1  and  3.  In  the 
case  of  conductor  No.  1  a  part  of  the  outgoing  current  returns 
along  conductor  No.  2  at  a  distance  d,  while  the  remainder 
returns  along  conductor  No.  3  at  a  distance  2d',  so  that  the  total 
flux  of  induction  surrounding  conductor  No.  1  must  necessarily 
be  greater  than  that  surrounding  conductor  No.  2.  The  same 
argument  applies  to  conductor  No.  3. 

Applying  formula  (3)  to  the  arrangement  of  conductors,  as 
shown  in  Fig.  2,  the  quantity  between  brackets  in  the  case  of 
conductor  No.  1  becomes: 

d       ,  ,      2d 
-  1 2  log  -  -  73  log  — 

=  -  (72  +  73)log^  -73log2 


364  ELECTRIC  POWER  TRANSMISSION 


The  total  induced  e.m.f  .  per  mile  of  conductor  No.  1  will  therefore 
be: 

El  =  a  X  [/i  log  ^  -  73  log  2]  +  67  j  (4) 

Similarly,  for  conductor  No.  3: 

E3  =  a  X  [/3  log  *  -  7i  log  2]  +  b/3  (5) 

while  the  volts  induced  in  the  middle  conductor  (No.  2)  will  be 
simply: 

E2  =  a  X  1  2  log  -  +  &72  (6) 

It  is  interesting  to  note  that  what  may  be  referred  to  as  the 
disturbing  element  in  the  case  of  the  two  outside  wires  (the 
quantities  73  log  2  and  I\  log  2  respectively)  is  not  dependent 
upon  the  actual  diameter  or  distance  apart  of  the  conductors. 
It  consists  of  an  e.m.f.  component  either  30  time-degrees  or  150 
time-degrees  behind  the  phase  of  the  line  current,  depending 
upon  the  order  of  the  phase  rotation;  and  the  magnitude  of  this 
e.m.f.  component  relatively  to  the  total  e.m.f.  of  self-induction 

will  depend  upon  the  value  of  the  ratio  -.     If  d  is  large  and 

r  relatively  small,  as  in  the  case  of  a  high-pressure  overhead 
transmission  system,  then  the  first  quantity  between  brackets, 
in  equations  (4)  and  (5),  is  relatively  large,  and  the  disturbing 
element  (/3  log  2  or  7i  log  2)  is  usually  negligible.  On  the  other 
hand,  if  the  conductors  consist  of  three  separate  single  cables, 
laid  side  by  side  in  a  trench,  with  the  distance,  d,  between  them 
small  in  comparison  with  the  diameter,  2r,  of  the  cables,  then  the 
"disturbing  element"  becomes  of  greater  importance  relatively 
to  the  total  induced  e.m.f. 

In  order  to  form  some  idea  of  the  magnitude  of  this  out-of- 
balance  component  of  the  induction,  it  will  be  well  to  work  out 
two  numerical  examples,  one  for  a  high-tension  overhead  scheme 
and  the  other  for  a  low-tension  transmission  system  with  the 
three  conductors  in  comparatively  close  proximity. 

Example  1.  —  Assumed  data:  Three-phase  power  transmitted 
=  20,000  kw.;  e.m.f.  =  110,000  volts;  power-factor  =  0.8;  fre- 
quency /  =  25  cycles  per  second;  length  of  line  =  200  miles. 
Conductors  of  aluminum;  diameter,  2r  =  0.6  in.  Minimum  dis- 


INDUCTANCE  OF  TRANSMISSION  LINES       365 

tance  between  -wires,  d  =  10  ft.  =  120  in.  On  the  above  data 
the  current  per  conductor  is  about  130  amp.  With  the  aid  of 
formulas  (4),  (5)  and  (6)  it  is  an  easy  matter  to  determine  the 
induced  e.m.fs.  in  the  several  conductors,  and  since  the  quantity, 

log  -  =  log  TT^-  =  2.6021,  while  log  2  =  0.3010,  it  will  at  once  be 

T  U.o 

seen  that  the  "disturbing  element"  is  relatively  small. 

The  e.m.fs.  induced  in  each  conductor  200  miles  long,  in  round 
figures  (neglecting  the  component  67  due  to  the  internal  flux) 
are  as  follows: 

In  the  middle  conductor  (No.  2),  8000  volts,  the  time-phase 
of  which  is  exactly  one-quarter  cycle  behind  the  time-phase  of 
the  current  /2. 

In  conductor  No.  1,  an  e.m.f.  component  of  8000  volts,  exactly 
a  quarter  cycle  behind  the  current  /i  less  another  component 
(referred  to  as  the  disturbing  element)  equal  to  about  915  volts, 
the  phase  of  which  is  exactly  one-quarter  cycle  behind  the 
current  1$.  The  resultant  is  the  difference  between  two  vector 
quantities  separated  by  a  time-phase  angle  of  120  deg.,  so  that 
this  resultant  is  actually  greater  than  either  of  the  two  compo- 
nents, as  will  be  shown  hereafter. 

In  conductor  No.  3  there  will  be  an  e.m.f.  component  of  8000 
volts,  one-quarter  cycle  behind  the  current  /a,  and  a  component 
of  915  volts,  one-quarter  cycle  behind  /i. 

Example  2.  —  Assumed  data:  Three-phase  power  transmitted 
=  20  kw.;  e.m.f.  =  110  volts;  power-factor  =  0.8;  frequency  / 
=  60  cycles  per  second  ;  current  per  wire  =  130  amp.  ;  distance  of 
transmission  =  %  mile;  three  single  cables  in  trench,  lying  in 
the  same  plane  with  a  distance  between  centers  d  =  3  in.; 
diameter  over  copper  =  2r  =  0.5  in. 

In  this  example  the  quantity  log  - 

3 


=  1.0792 
The  ratio  between  log  2  and  this  number  is  =  °-28- 


is  to  say,  the  component  of  the  total  induced  e.m.f.,  which  appears 
only  in  the  two  outside  conductors,  as  indicated  by  formulas 
(4)  and  (5)  is,  in  this  example,  numerically  greater  than  a  quarter 
of  the  more  important  component;  while  in  the  previous  example 
of  a  high-tension  overhead  transmission  system  the  ratio  was 


366 


ELECTRIC  POWER  TRANSMISSION 


0.115,  being  considerably  smaller  because  of  the  greater 


0.30JO 
2.6021 
distance  between  the  wires. 

Vector  Diagram  Illustrating  Example  2. — The  vectors  I\,  2z 
and  73  in  Fig.  3  represent  the  currents  in  the  three  conductors, 
the  time-phase  angle  between  them  being  120  deg.  The  rotation 
of  the  phases  is  assumed  to  be  in  the  order  7i,  72,  Is',  in  other 
words,  72  lags  behind  7i  by  one-third  of  a  cycle,  and  73  lags  behind 
7  2  also  by  one-third  of  a  cycle.  The  lengths  of  these  vectors  are 
such  as  to  represent  the  line  current  of  130  amp.;  but,  as  the 
diagram  has  been  drawn  to  illustrate  the  phase  angles  and  magni- 
tudes of  the  various  components  of  the  induced  e.m.fs.,  the  magni- 
tude of  the  current  vectors  need  not  be  considered.  If  the  numer- 
ical values  of  the  induced  volts 
are  determined  with  the  aid  of 
formulas  (4),  (5)  and  (6),  it  will 
be  found  that  the  component 
common  to  all  three  conductors 
amounts  to  21.6  volts,  while  the 
"disturbing  element" — that  is, 
the  component  appearing  in  the 
two  outer  conductors  only — 
amounts  to  5.5  volts. 

The  vectors  OB,  OVZ  and  OD 
must,  therefore,  be  drawn  of 
such  a  length  as  to  represent 
21.6  volts  in  a  direction  exactly 
90  time-degrees  behind  the  cor- 
responding current  vectors;  and,  so  far  as  the  middle  conductor 
is  concerned,  the  vector  OV2  will  represent  the  whole  of  the  in- 
duced e.m.f.;  but  in  the  case  of  conductor  No.  1  (carrying  current 
7i),  OA  must  be  drawn  exactly  90  time-degrees  in  advance  of  07  3 
— that  is,  exactly  opposite  to  07),  because  of  the  negative  sign 
in  equation  (4) — and  of  such  a  length  as  to  represent  5.5  volts. 
By  combining  OA  with  OB  in  the  usual  way,  OVi  is  obtained  as 
representing  the  total  e.m.f.  induced  in  conductor  No.  1.  In  a 
similar  manner  OV3  is  obtained  for  the  total  induced  e.m.f.  in 
conductor  No.  3.  It  is  interesting  to  note  that  OVi  lags  be- 
hind the  current  7i  by  a  time  interval  greater  than  a  quarter 
period,  while  the  lag  of  the  induced  volts  F3  behind  the  current 
7  3  is  less  than  a  quarter  period. 


B 


FIG.    3. — Vector    diagram.      Three 
conductors  in  same  plane. 


INDUCTANCE  OF  TRANSMISSION  LINES       367 


In  the  particular  example  under  consideration  the  calculated 
value  of  Fi  or  F3  is  24.8  volts;  F2  being  21.6  volts. 

It  is  not  difficult  to  understand  why  the  magnitude  and  phase 
relations  of  the  induced  e.m.fs.  in  the  various  conductors  of  a 
polyphase  transmission  are  not  the  same  for  an  unsymmetrical 
arrangement  of  conductors  as  for  an  arrangement  in  which  each 
conductor  is  similarly  placed  in  relation  to  all  the  other  conduc- 
tors. With  an  unsymmetrical  arrangement,  the  unbalancing 
effect  may  be  said  to  be  due  to  the  mutual  induction  between  the 
loops  formed  by  different  pairs  of  wires;  there  may,  in  fact,  be 
a  transfer  of  energy  between  one  loop  and  another  just  as  in  the 
case  of  the  primary  and  second- 
ary windings  of  a  transformer. 

Effect  of  Transposing  the 
Conductors. — If  each  conductor 
of  the  arrangement  referred  to 
in  the  above  example  is  made 
to  occupy,  in  turn,  the  position 
midway  between  the  remaining 
two  conductors  for  a  distance 


FIG.  4. — Vector  diagram  illustrat- 
ing effect  of  transposing  conductors 
lying  in  the  same  plane. 


equal  to  one-third  of  the  total 
distance  of  transmission,  it  is 
obvious  that  the  out-of-balance 
effect  will  be  corrected.  It  will, 
however,  be  of  interest  to  as- 
certain what  will  be  the  numeri- 
cal value  of  the  (equal)  voltages  induced  in  the  three  conductors 
if  transposed  in  the  manner  suggested.  It  is  not  necessary  to 
consider  more  than  one  of  the  conductors,  and,  in  Fig.  4,  OB  repre- 
sents (as  in  Fig.  3)  that  portion  of  the  e.m.f.  induced  in  conductor 
No.  1  which  remains  unaltered  whether  the  conductor  be  midway 
between  the  other  two,  or  be  itself  one  of  the  outer  conductors. 
The  length  of  this  vector  will,  therefore,  be  such  as  to  represent 
21.6  volts.  Now,  when  the  arrangement  of  the  conductors  is  in 
the  order  1,2,3  (as  in  Fig.  2),  the  "disturbing  element "  will  be  BG, 
drawn  90  degrees  in  advance  of  OI3,  exactly  as  OA  (or  BVi)  in  Fig. 
3 ;  but  the  length  of  this  vector,  instead  of  being  equivalent  to  5.5 
volts,  will  be  only  one-third  of  this  value,  or  1.83  volts,  because 
conductor  No.  1  occupies  this  position  over  one-third  of  the 
total  distance  of  transmission.  When  the  arrangement  of  the 
conductors  is  1,  3,  2,  the  "disturbing  element"  will  be  GVi 


368  ELECTRIC  POWER  TRANSMISSION 

(Fig.  4),  drawn  90  degrees  in  advance  of  072.  Clearly  BGVi 
is  an  equilateral  triangle,  and  the  resultant  of  the  induced  e.m.f. 
in  conductor  No.  1  is  OVi,  drawn  90  time-degrees  behind  the 
current  vector  01  1  and  equal  in  magnitude  to  the  algebraic  sum 
of  OB  =  21.6  volts  and  BVi  =  one-third  of  5.5  volts. 

If,  therefore,  the  wires  of  a  transmission  line  are  disposed  in 
one  plane,  as  indicated  in  Fig.  2,  but  transposed  at  intervals  so 
that  each  wire  shall  occupy  the  middle  position  over  a  space 
equal  to  one-third  of  the  distance  of  transmission,  then  the 
resultant  induced  e.m.f.  per  conductor  will,  so  far  as  phase  is 
concerned,  lag  behind  the  current  by  a  quarter  period,  exactly 
as  if  the  wires  occupied  the  vertices  of  an  equilateral  triangle; 
but  the  amount  of  the  induced  volts  will  be  somewhat  greater 
than  in  the  latter  case,  under  otherwise  similar  conditions. 

The  numerical  value  of  the  induced  volts  per  conductor  — 
that  is,  the  length  of  the  vector  OVi  in  Fig.  4  —  can  be  calculated 
by  the  formula: 

W  (7) 


where  /  is  the  current  in  any  one  conductor,  and  the  two  quan- 
tities between  brackets  have  merely  to  be  added  algebraically. 
If  preferred  the  quantity  between  brackets  may  be  written: 


or     7  log  (l.26—  J,  so  that  formula  (7)  appears  in  the  form: 

E  =  0.00466/7  log  ^^  +  0.000506/7  (8) 

Inductance  of  Electric  Transmission  Lines  as  Affected  by 
the  Subdivision  of  the  Circuits  and  the  Arrangement  of  the 
Conductors.  —  There  are  reasons  in  favor  of  transmitting  large 
amounts  of  electric  power  through  two  or  more  sets  of  wires, 
quite  distinct  from  mechanical  considerations  or  the  increased 
security  against  a  total  shut-down  in  the  event  of  accidents. 
The  inductive  drop  of  pressure  may  be  reduced  by  substituting, 
for  a  single  set  of  transmission  lines,  two  or  more  sets  of  suitably 
arranged  lines  of  a  correspondingly  reduced  cross-sectional 
area.  Whether  or  not  the  subdivision  of  a  transmission  line 
into  two  or  more  parallel  circuits  would  be  justifiable  in  practice 
will  depend  upon  economic  and  other  considerations  which  it  is 
not  proposed  to  touch  upon  here. 


INDUCTANCE  OF  TRANSMISSION  LINES       369 

Single-phase  Systems.  —  In  Fig.  5  the  two  conductors  of  a 
single-phase  transmission  are  shown,  with  distance  d  between 
centers  of  wires.  The  current  may  be  considered  as  going  out 
through  the  conductor  1  and  returning  through  conductor  2. 
The  diameter  of  the  wire  is  assumed  to  be  2r  and  the  current  / 
amp. 

The  formula  which  gives  the  induced  volts  per  mile  of  single 
conductor  when  the  whole  of  the  current  may  be  considered  as 
returning  at  a  distance  d  from  the  center  of  the  outgoing  con- 
ductor is 

(9) 


where  a  has  the  value  given  above,  and  the  item  67  has  been 

omitted  as  it  is  not  necessary  to  include  it  when  considering 

differences  of  reactive  e.m.fs.,  especially  in  the  case  of  overhead 

systems  where  its  magnitude  is  relatively 

small,    and    frequently    negligible.      This 

formula  alone  is  sufficient  to  indicate  that 

an  improvement  in  the  matter  of  indue- 

tive  voltage  drop  is  to  be  expected  if,  in- 

stead  of  transmitting  the  total  current  / 

through  one  pair  of  conductors,  there  be  provided  two  or  three 

pairs  of  conductors  spaced  sufficiently  far  apart  to  prevent  mutual 

inductive  effects,  each  pair  being  of  sufficient  cross-section  to  carry 

one-half  or  one-third  of  the  total  current,  as  the  case  may  be;  be- 

cause, although  the  quantity  log-  will  increase  slightly  on  account 

of  the  reduction  in  the  dimension  r,  this  increase  will  not  be  of 
nearly  so  much  importance  as  the  reduction  of  /. 

Numerical  Example.  —  In  order  to  illustrate  the  above  point 
a  few  examples  will  be  worked  out  based  on  the  following  as- 
sumed data: 

Total  current,  /  =  100  amp. 

Diameter  of  single  conductor  to  transmit  the  total  current, 
2r  =  0.5  in. 

Frequency,  /  =  60  cycles,  from  which  a  =  0.279. 

Distance  between  centers  of  wires  (corresponding  to  a  pres- 
sure of  about  50,000  volts),  d  =  70  in. 

If  the  transmission  line  is  divided  into  two  equal  sections, 
the  current  in  each  section  will  be  50  amp.,  and  for  equal  total 
weight  of  copper  (leading  to  the  same  ohmic  drop  of  pressure), 

24 


370  ELECTRIC  POWER  TRANSMISSION 

the  radius  of  each  conductor  will  be  r  -r-  \/2.  Similarly,  if  there 
are  three  equal  sections,  the  current  will  be  33.33  amp.,  and  the 
radius  of  the  conductors  r  -4-  \/3. 

The  induced  volts  as  given  by  formula  (9)  work  out  as  follows 
for  the  three  conditions : 

Single  pair  of  lines 

e  =  68.34  volts  (10) 

Two  pair  of  lines  of  equal  total  cross-section, 

e  =  36.25  (11) 

Three  pair  of  lines  of  equal  total  cross-section, 

e  =  25.00  (12) 

These  figures  show  that  the  inductive  drop  of  pressure  on  a 
single-phase  transmission  may  be  reduced  by  splitting  up  the 
current  and  transmitting  along  two  or  more  pairs  of  lines  spaced 
sufficiently  far  apart  to  prevent  appreciable  magnetic  interfer- 
ence between  the  sets  of  lines;  and  the  reduction  of  the  inductive 
drop  is  very  nearly  in  proportion  to  the  number  of  subdivisions 
of  the  single  line. 

Although  electric  transmission  systems  have  been  arranged 
with  two  distinct  sets  of  conductors  run  upon  separate  pole 
lines  spaced  sufficiently  far  apart  to  avoid  magnetic  interfer- 
ence, such  an  arrangement  is  necessarily  costly.  Consider, 
therefore,  two  alternative  arrangements,  shown  in  Figs.  6  and  7, 
by  which  a  single  circuit  can  be  split  up  into  two  parallel  cir- 
cuits, the  four  wires  being  carried  on  the  one  set  of  poles  with 
the  spacing  between  the  individual  wires  as  small  as  possible — 
that  is,  such  that  in  no  case  shall  the  distance  d  between  out- 
going and  return  conductors  be  less  than  the  minimum  deter- 
mined by  the  voltage  of  the  supply. 

In  Fig.  6  is  shown  a  symmetrical  arrangement  with  the  four 
conductors  of  equal  cross-section  occupying  the  corners  of  a 
square;  the  outgoing  conductors  are  marked  1  and  3,  and  the 
return  conductors,  2  and  4.  Even  if  the  two  circuits  1-2  and 
3-4  are  connected  in  parallel  at  both  ends  of  the  line,  the  sym- 
metry of  the  arrangement  will  insure  that  the  total  current 
will  divide  itself  equally  between  the  two  sets  of  conductors. 
The  effective  or  resultant  magnetic  flux  surrounding  any  one 
conductor  will,  for  the  same  reason,  be  equal  to  that  which 


INDUCTANCE  OF  TRANSMISSION  LINES       371 

surrounds  any  one  of  the  remaining  three  conductors.  It  will, 
therefore,  suffice  to  calculate  the  e.m.f.  of  self-induction  gen- 
erated in  any  one  conductor. 

Consider  the  conductor  1,  in  which  there  is  the  current  ~- 

If  the  other  outgoing  conductor,  3,  were  situated  anywhere  on 
the  dotted  circle  of  radius  d,  passing  through  2  and  4,  then  the 
magnetic  effect  of  the  current  in  3 — so  far  as  conductor  1  is  con- 
cerned— would  counteract  the  effect  of  the  return  current  in 
either  2  or  4.  On  the  basis  of  the  data  previously  assumed,  the 
flux  around  1  would  generate  an  e.m.f.  of  36.25  volts,  as  in  equa- 
tion (11).  If,  on  the  other  hand,  conductor  3  were  coincident  with 
1,  there  would  be  the  condition  of  the  full  current  I  in  the  conduc- 


-* 

••*,     v-    -i 

FIG.  6.  FIG.  7. 

FIGS.  6  and  7. — Alternative  arrangements  of  conductors.     Single-phase 
transmission. 

tor  1,  the  whole  of  which  would  be  returning  at  a  distance  d, 
and  the  induced  volts  would  be  68.34,  as  given  in  equation  (10). 
With  the  conductor  3  situated  at  a  distance  \/2d  from  conduc- 
tor 1,  as  shown  in  Fig.  6,  the  resultant  effective  flux  surrounding 
conductor  1  may  be  considered  as  the  difference  between  the 
flux  due  to  a  current  I  up  to  a  distance  d  less  the  flux  due  to  a 
current  1/2  up  to  a  distance  -\/2d;  and  this  resultant  flux  would 
produce  a  back  e.m.f. 

E~ai  log  -^—fr  -  4logr^v72  <13) 

On  the  data  previously  assumed,  the  e.m.f.  is 

E  =  72.5  -  38.39  =  34.11  volts.  (14) 

Thus,  by  arranging  the  conductors  of  the  divided  circuit  in  the 
manner  shown  in  Fig.  6,  which  permits  of  the  four  wires  being 


372  ELECTRIC  POWER  TRANSMISSION 

supported  on  the  one  set  of  poles,  a  better  result  is  obtained  in 
regard  to  inductive  voltage  drop  than  if  the  two  circuits  had 
been  run  entirely  separately;  the  voltage  drop  in  this  latter 
case  being  36.25,  as  in  equation  (11). 

If,  on  the  other  hand,  the  position  of  one  pair  of  conductors 
be  assumed  to  be  reversed,  as  indicated  in  Fig.  7,  then  the 
magnetic  flux  in  the  loop  formed  by  the  outgoing  and  return 
conductors  2  and  3  has  no  effect  on  the  conductors  1  and  4, 
and  the  effective  flux  surrounding  any  one  conductor  is  clearly 

that  due  to  a  current  ~  returning  at  a  distance  -\/2d:  the  in- 
duced volts  per  conductor  will  be  38.39,  this  being  the  value 
of  the  second  term  in  formula  (13). 
With   an  arrangement  of  conduc- 
tors, as  in  Fig.  7,    it   is  obvious 
\^        that  the  conditions  are  worse  than 
<       %,        if  the  two  circuits  are  quite  dis- 
^Ax  tinct,  because  a  portion  of  the  flux 
— J — -^  /  3'    produced  by  one  pair  of   conduc- 
'        tors,  such  as  3  and  4,  passes  also 
through  the  loop   1-2,  thereby  in- 
creasing the  inductive  drop  in  these 
wires. 
FIQ.  8.— Arrangement   of  con-        Three-phase  Systems. — The  satis- 

ductors — t  h  r  e  e  -  phase    transmis- 


to  inductive  drop  when  a  single- 
phase  circuit  is  split  up  into  two  circuits  arranged  as  indicated 
in  Fig.  6,  suggest  that  a  somewhat  similar  arrangement  might 
be  adopted  with  advantage  in  the  case  of  polyphase  transmissions. 
An  arrangement  of  wires  suitable  for  three-phase  transmission 
is  shown  in  Fig.  8.  Here  the  three-phase  line  is  supposed  to  be 
split  up  into  two  parallel  three-phase  circuits,  1,  2,  3  and  1',  2', 
3'.  The  arrangement  being  symmetrical  and  all  conductors 
being  assumed  to  be  of  equal  size,  the  same  amount  of  current 
will  be  carried  by  each  of  the  six  conductors,  provided  the  load  is 
a  balanced  one,  such  as  is  usual  in  the  case  of  a  three-wire, 
three-phase  system. 

With  the  arrangement  of  wires  as  in  Fig.  8  the  minimum  dis- 
tance d  is  maintained  between  all  wires  at  different  potentials, 
and  the  current  in  conductors,  such  as  1  and  I',  placed  at  opposite 


INDUCTANCE  OF  TRANSMISSION  LINES       373 

ends  of  a  diameter,  will  be  of  the  same  time-phase  and  equal  in 
magnitude. 

It  will  be  interesting  to  work  out  a  numerical  example  based  on 
data  already  assumed  in  connection  with  the  single-phase  trans- 
mission, namely,  a  current  of  100  amp.  per  phase  and  a  mini- 
mum distance,  d,  of  70  in.  between  conductors  at  different  poten- 
tials. The  points  to  bear  in  mind  are: 

(1)  That  owing  to  the  symmetrical  arrangement  of  the  con- 
ductors, with  the  rotation  of  the  phases  always  in  the  same  direc- 
tion, the  total  effective  magnetic  flux  around  any  one  conductor  is 
the  same  (except  in  regard  to  phase)  as  that  which  surrounds  any 
one  of  the  other  five  conductors.     The  calculations  can  therefore 
be  made  for  any  one  conductor,  such  as  No.  1. 

(2)  That  the  current  in  any  outgoing  conductor,  such  as  1,  may 
be  considered  as  returning  through  the  five  remaining  conductors, 
due  attention  being  paid  to  phase  relations. 

(3)  That  the  resultant  of  the  currents  in  conductors  2'  and  3', 
or  the  resultant  of  the  currents  in  conductors  2  and  3,  is  equivalent 
to  a  current  equal  to  that  in  conductor  1,  but  of  opposite  phase. 
The  total  effective  flux  around  conductor  1  may,  therefore,  be 
considered  as  the  resultant  of  three  component  fluxes: 

(a)  A  flux  due  to  a  current  ~  returning  (through  2'  — 3')  at  a 
distance  d;  plus  (b)  a  flux  due  to  a  current  ^  returning  (through 

2-3)  at  a  distance  \/3d;  less  (c)  a  flux  due  to  a  current  ^  re- 
turning (through  1')  at  a  distance  2d. 

The  numerical  values  for  the  induced  volts  are  found  to  be: 

(a)  =  36.25  [being  the  same  as  inequation  (11)];   (6)  =  39.47; 
(c)  =  40.49;  and  (a)  -f  (b)  -  (c)  =  35.23. 

If  two  separate  three-phase  lines  spaced  a  considerable  distance 
apart  were  substituted  for  the  arrangement  in  Fig.  8,  the  induced 
volts  per  mile  per  conductor  would  be  as  given  in  equation  (11), 
namely,  36.25,  assuming  the  triangular  arrangement  of  wires,  with 
distance  d  between  them.  The  arrangement  shown  in  Fig.  8  is 
therefore  slightly  better  from  the  point  of  view  of  inductive  drop, 
notwithstanding  that  both  sets  of  wires  can  be  run  on  the  same 
pole  line  with  no  greater  spacing  between  wires  than  the  minimum 
distance  d  determined  by  the  voltage  between  phases.  The  fig- 


374  ELECTRIC  POWER  TRANSMISSION 

ure  35.23  volts  for  the  split  three-phase  system  may  be  compared 
with  34.11  volts  as  given  in  equation  (14)  relating  to  the  single- 
phase  transmission  with  two  circuits.  It  is  clear  that  in  either 
example,  the  drop  in  volts  per  conductor  in  the  undivided  circuit, 
with  each  conductor  of  sufficient  section  to  carry  the  total  current 
of  100  amp.,  would  be  68.34,  as  given  by  equation  (10). 


APPENDIX  II 
SPECIFICATION  FOR  WOOD  POLE  TRANSMISSION  LINE 

1.  General  Description  of  Transmission  Line. — This  transmis- 
sion line,  which  is  25  miles  long,  connects  the  water  power  gen- 
erating station  at in  the  mining  district  of with 

the  substation  at  the  mines.     The  system  will   be  three  phase 
with   a  pressure  of  22,000  volts  between  wires  supported  on 
wood  poles.     The  conductors  will  be  No.  2/0  stranded  alumi- 
num.    The  average  span  will  be  150  ft.,  and  the  separation 
between  wires  will  be  3  ft.,  the  three  conductors  being  arranged 
in  the  form  of  an  equilateral  triangle  with  one  conductor  at  the 
top  and  the  remaining  two  conductors  below,  at  the  ends  of  a 
wooden  cross-arm,  all  as  shown  on  drawing  No 

Where  long  spans  are  necessary,  a  double-pole  arrangement, 

as  shown  on  drawing  No ,  will  be  adopted.  Particulars 

relating  to  special  precautions  and  methods  of  procedure  in  the 
case  of  exceptionally  long  spans,  will  be  found  in  clause  (8)  under 
the  heading  "Spans." 

There  will  be  no  telephone  wires  supported  on  the  transmission- 
line  poles. 

There  will  be  no  grounded  guard  wire  above  the  conductors, 

but  galvanized  iron  lightning  rods,  as  shown  on  drawing  No , 

will  be  fitted  to  every  third  pole  on  the  average.  Further  par- 
ticulars relating  to  protection  against  lightning  are  given  in  clause 
(7)  under  heading  "Grounding."  For  particulars  of  sags  and 
tensions,  refer  to  clause  (12)  under  heading  "Stringing  of  Wires." 

2.  Clearing.— The  width  of  the  right-of-way  shall  be  100  ft., 
and  all  lumber,  brush  and  other  growth  of  every  description 
must  be  cut  and  cleared  so  that,  in  no  portion  of  the  right-of- 
way,  shall  the  tops  of  tree  stumps,  undergrowth  or  bush  be 
higher  than  18  inches  above  ground  level. 

At  all  points  where  a  space  of  50  ft.  on  each  side  of  the  pole 
line  is  insufficient  to  prevent  possible  damage  to  wires  by  falling 
trees,  the  normal  width  of  the  clearing  must  be  exceeded. 

All  useful  material  shall  be  separated  and  suitably  stacked  at 
a  safe  distance  from  waste  material  piled  for  burning. 

375 


376  ELECTRIC  POWER  TRANSMISSION 

All  tops,  limbs,  brush  and  other  waste  shall  be  burned,  great 
care  being  taken  to  prevent  spread  of  fire  beyond  the  limits  of 
clearing.  Suitable  fire-fighting  appliances  shall  be  kept  at  hand 
while  burning  is  proceeding. 

3.  Poles. — Cedar  poles  shall  be  used  when  obtainable;  but, 
owing  to  difficulties  of  transport,  it  is  proposed  to  make  use  of  the 
poles  (mainly  pine)  obtained  in  the  neighborhood  of  the  trans- 
mission line  while  clearing  the  right-of-way. 

Dimensions. — The  greater  number  of  the  poles  required  will 
be  35  ft.  long:  they  shall  be  sawn  square  at  both  ends.  These 
poles  shall  measure  not  less  than  24  in.  in  circumference  at  the 
top  under  bark,  and  not  less  than  38  in.  under  bark  6  ft.  from 
butt.  The  approximate  number  required  will  be  840.  In 
addition  to  these,  about  100  poles  45  ft.  long  will  be  required, 
and  these  shall  measure  not  less  than  24  in.  in  circumference  at 
top  and  42  in.  6  ft.  from  butt. 

Quality. — All  poles  to  be  cut  of  best  quality  live  green  timber, 
well  proportioned  from  butt  to  top  and  well  seasoned;  the  bark 
to  be  peeled,  and  all  knots  and  limbs  closely  trimmed.  The 
poles  shall  be  reasonably  straight,  and  no  poles  having  short 
crooks  or  a  reverse  curve  will  be  accepted.  The  amount  of 
"sweep"  measured  between  six-foot  mark  and  top  of  pole  shall 
not  exceed  8  in.  in  the  35-ft.  poles,  or  11  in.  in  the  45-ft.  poles. 

Twisted  Poles. — No  poles  having  more  than  two  complete 
twists  in  the  total  length,  and  no  cracked  poles  will  be  accepted. 

Dead  Poles. — No  dead  poles  or  poles  having  dead  streaks 
covering  more  than  one-quarter  of  their  surface  will  be  accepted. 

Butt  Rot. — This  must  not  exceed  10  per  cent,  of  the  cross-sec- 
tion of  the  pole,  and  the  diameters  of  poles  with  butt  rot  or  hollow 
hearts  must  be  substantially  greater  than  the  corresponding 
diameters  of  sound  poles.  Poles  with  hollow  hearts  exceeding 
8  in.  in  diameter  will  not  be  accepted.  If  average  diameter  of 
rot  does  not  exceed  6  in.,  the  butt  measurement  must  be  2  in. 
greater  than  in  the  case  of  sound  poles.  If  the  average  diameter 
of  rot  is  7  in.,  the  butt  measurement  must  be  4  in.  greater. 

Miscellaneous  Defects. — Poles  with  sap  rot,  woodpeckers' 
holes,  plugged  holes,  also  poles  that  have  been  attacked  by  ants, 
worms,  or  grubs,  are  liable  to  be  rejected  as  unsuitable. 

The  treatment  of  all  poles  before  erection  shall  be  as  follows: 
The  gains  shall  be  sawn  square  with  the  axis  of  the  pole  and  in 
such  a  position  that,  when  erected,  the  curvature  of  the  pole 


WOOD  POLE  TRANSMISSION  LINES  377 

(if  any)  shall  be  in  the  direction  of  the  line.     The  position  of  the 

gains  is  indicated  on  the  accompanying  drawing  No 

showing  the  standard  pole  construction.  The  gains  shall  be  not 
less  than  %  in.  and  not  more  than  %  in.  deep;  they  shall  be 
accurately  cut  so  that  the  cross-arms  will  have  a  driving  fit,  and 
the  holes  for  the  %-in.  bolts  securing  cross-arm  to  pole  shall  be 
bored  after  the  cross-arm  has  been  fitted  in  position.  These 
holes,  together  with  all  other  necessary  holes,  as  indicated  on 
drawings,  shall  be  bored  clean  and  true  without  splintering. 
The  holes  for  lag  screws  securing  braces  to  poles  shall  be  bored 
after  braces  have  been  fitted  to  cross-arms;  they  must  be  small 
enough  in  diameter  to  ensure  that  the  threads  of  the  lag-screw 
shall  engage  properly  in  the  wood. 

The  butts  of  all  poles,  together  with  the  gains  and  tops,  shall 
be  treated  with  two  coats  of  coal-tar-creosote  oil,  heated  to 
about  220°  F.  and  applied  with  a  brush.  At  least  24  hours  must 
be  allowed  to  elapse  between  applications.  The  painting  of 
the  butts  shall  be  carried  at  least  18  inches  above  ground  level. 

4.  Cross-arms. — The    cross-arms   shall   be   of  yellow   birch, 
Oregon  fir,  or  long-leaf  yellow  pine,  well  seasoned,  close  grained, 
and  free  from  knots  or  sap  wood*.     They  must  be  dressed  on  all 
sides.     They  must  measure  4J^  in.  deep  by  3^  in.  wide,  and 

be  bored,  as  indicated  on  drawing  No ,  with  templet,  true 

and   symmetrical:  the  holes  to  be  bored   clean   and   without 
splintering.     After  having  been  bored,  the  cross-arms  shall  be 
painted  with  two  coats  of  good  asphaltum  paint.     In  cases 
where  double  cross-arms  are  required,  it  will  be  necessary  to  bore 
the  standard  cross-arms  with  additional  holes  for  the  %-in. 
spacing  bolts,  the  position  of  which  is  shown  on  the  pole  drawings 
previously  referred  to. 

5.  Grading. — An  effort  should  be  made  to  maintain  as  far  as 
practicable  an  even  grade.     By  carefully  choosing  the  location 
of  each  pole  so  as  to  avoid  the  highest  points  and  greatest  depres- 
sions when  passing  over  uneven  ground,  it  may  be  possible  to 
avoid  the  use  of  poles  differing  in  length  to  any  great  extent. 
Should  it  be  necessary  to  shorten  a  pole,  this  must  be  done  by 
sawing  a  piece  off  the  butt  end;  but  unless  this  is  done  before  the 
treatment  with  preservative  liquid,  the  butt   must   receive   a 
further  treatment  with  the  creosote  oil  before  erection  of  the  pole. 
In  some  cases  where  the  ground  is  favorable,  the   shortening 
of  poles  may  be  avoided  by  digging  the  hole  deeper  than  would 


378  ELECTRIC  POWER  TRANSMISSION 

otherwise  be  necessary.  When  using  shortened  poles,  and  when 
passing  over  uneven  ground,  it  is  important  to  bear  in  mind 
that  under  no  condition  shall  the  bottom  conductors  hang 
lower  than  18  ft.  above  the  ground,  and  when  crossing  tote 
roads  or  public  footpaths,  the  minimum  distance  between  wire 
and  ground  shall  be  21  ft. 

6.  Pole  Setting. — Where  poles  are  set  in  good  solid  ground,  the 
depth  of  holes  shall  be  as  follows: 

35-ft.  poles  on  straight  runs 5%  ft. 

45-ft.  poles  on  straight  runs 6      ft. 

35-ft.  poles  at  corners  or  where  stresses  are  excessive .  .  6      ft. 

45-ft.  poles  at  corners  or  where  stresses  are  excessive .  .  6%  ft. 

If  the  ground  is  soft,  the  depth  of  setting  shall  be  6  in.  greater 
than  when  setting  in  solid  ground.  If  the  soil  is  very  soft,  but  not 
such  as  would  be  described  as  swampy,  one  or  more  transverse 
logs  may  be  bolted  to  the  butt  of  the  pole  in  order  to  obtain  addi- 
tional bearing  area. 

When  erected  in  solid  rock,  the  depth  of  hole  shall  not  be  less 
than  3%  ft. 

In  loose  or  sandy  soil,  the  sand  barrel  or  its  equivalent  should 
be  used.  This  must  be  filled  with  a  firm  soil  which  may  contain 
stone  or  rock. 

In  swampy  ground  the  base  of  the  pole  must  be  provided  with 
an  arrangement  of  transverse  timbers  securely  braced  to  the  pole, 
in  addition  to  which  the  hole  shall,  if  necessary,  be  lined  with 
sheet  piling  and  filled  with  good  soil  which  may  contain  stones  or 
rock.  As  an  alternative,  a  stone-  or  rock-filled  crib  may  be  built 
round  the  butt  of  the  pole  above  ground  level.  In  some  cases 
concrete  may  be  used  with  advantage  in  the  pole  foundation,  but 
it  will  generally  be  found  that  the  use  of  concrete  can  be  avoided. 

Poles  must  not  be  set  along  the  edge  of  cuts  or  embankments 
or  where  the  soil  is  liable  to  be  washed  out,  unless  special  precau- 
tions are  taken  to  ensure  durable  foundations. 

When  setting  the  poles  in  good  ground,  the  holes  shall  be  dug 
of  ample  size  to  allow  of  easy  entrance  of  the  butts,  and  the  size 
at  bottom  must  be  large  enough  to  admit  of  the  proper  use  of 
tampers.  When  back-filling  holes,  there  should  be  not  less  than 
three  tampers  to  one  shoveller,  in  order  to  ensure  that  the  dirt 
shall  be  packed  tight.  In  no  case  must  the  earth  be  thrown  in  to 
a  greater  depth  than  6  inches  without  being  tamped  hard  before 


WOOD  POLE  TRANSMISSION  LINES  379 

the  next  layer  is  thrown  in.  The  proper  filling  of  holes  is  a  matter 
of  great  importance.  When  the  filling  is  properly  done,  it  should 
not  be  necessary  to  remove  any  excess  soil ;  this  should  be  packed 
firmly  around  the  pole,  the  object  being  to  raise  the  level  of  the 
ground  near  the  pole  and  so  cause  water  to  drain  from,  rather 
than  toward,  the  butt. 

When  setting  poles  on  a  straight  run,  the  lining  up  should  be 
done  with  a  transit,  and  the  poles  placed  with  cross-arms  truly  at 
right  angles  to  the  direction  of  the  line.  Where  the  direction  of 
the  line  alters,  the  poles  at  the  angles  must  be  set  so  that  the  cross- 
arm  halves  the  angle.  If  the  deviation  exceeds  5  degrees,  the 
corner  poles  shall  be  provided  with  double  cross-arms  and  fixtures. 
When  possible,  the  cross-arms,  braces,  and  other  fixtures  (but  not 
the  insulators)  should  be  mounted  on  the  poles  before  erection. 

7.  Grounding. — The  proper  grounding  of  lightning  rods  on 
the  pole  line  is  a  matter  of  importance.  Judgment  must  be  used 
in  determining  when  and  how  to  ground  the  poles;  but  either  of 
the  following  alternative  methods  will  be  considered  satisfactory, 
provided  the  soil  is  reasonably  moist: 

(1)  A  piece  of  galvanized  iron  pipe  1^  in.  in  diameter  and  8 
or  9  ft.  long  shall  be  buried  in  the  hole  alongside  the  butt  or  driven 
into  soft  soil,  the  ground  wire  being  attached  thereto  in  such  a 
manner  as  to  ensure  a  good  and  enduring  electrical  contact. 

(2)  The  ground  wire,  consisting  of  %$  in.  galvanized  stranded 
steel  cable,  after  being  carried  straight  down  the  side  of  the  pole 
and  secured  with  cleats,  shall  be  wound  spirally  around  the  butt 
and  carried  right  down  to  the  bottom  of  the  pole.     Not  more  than 
15  ft.  of  wire  should  be  buried  in  the  ground. 

It  is  of  little  use  to  ground  a  pole  in  solid  rock,  but  where  a  pole 
is  set  in  rock,  it  may  be  found  that  the  ground  wire  can  be  carried 
down  the  face  of  the  rock,  or  in  a  crevice,  to  a  point  where  a 
good  ground  can  be  obtained.  Where  grass  is  growing,  the  soil 
will  usually  contain  sufficient  moisture  to  afford  a  reasonably 
good  ground.  When  the  ground  wire  does  not  enter  the  ground 
alongside  the  pole,  sudden  bends  or  turns  should  be  avoided  in 
the  wire  connecting  the  lightning  rod  with  ground  plate  or  pipe. 

It  is  not  intended  to  provide  all  poles  with  lightning  rods; 
but,  except  when  the  soil  is  clearly  unsuitable  for  a  ground  con- 
nection, the  poles  in  the  positions  described  below  shall  be 
grounded : 


380  ELECTRIC  POWER  TRANSMISSION 

Both  poles  supporting  extra  long  spans  requiring  the  double 

pole  arrangement  as  shown  on  drawing  No previously 

referred  to. 

The  poles  on  each  side  of  railway  crossings. 

All  guyed  poles. 

The  six  poles  nearest  to  generating  station. 

The  six  poles  nearest  to  substation. 

In  addition  to  the  above-mentioned  poles,  one  pole  out  of 
every  three  poles  shall  be  grounded.  It  is  not  necessary  that 
every  third  pole  be  grounded:  judgment  must  be  used  in  deter- 
mining the  location  of  the  poles  to  be  grounded.  As  a  general 
rule,  it  is  more  important  to  ground  poles  on  heights  and  in 
exposed  positions  than  those  on  the  lower  ground;  but,  on  the 
other  hand,  it  is  of  little  advantage  to  ground  where  the  soil  is 
dry  or  otherwise  unsuitable.  In  exposed  positions  it  may  be 
advisable  to  ground  two  or  more  consecutive  poles,  while  in  unex- 
posed  positions  four  or  five  consecutive  poles  may  be  left  without 
lightning  rods. 

8.  Spans. — The  standard  length  of  span  shall  be  150  ft. 
Shorter  spans  must  be  used  at  angles  and  on  curves,  as  mentioned 
in  clause  9.  If  the  span  exceeds  1 70  ft.  the  poles  must  be  specially 
selected  for  strength.  No  span  greater  than  190  ft.  shall  be 
carried  on  single  poles.  For  longer  spans,  the  double-pole 

arrangement  as  shown  on  drawing  No previously  referred 

to,  shall  be  adopted,  with  a  horizontal  spacing  of  5  ft.  between 
wires  for  spans  up  to  600  ft. ;  but  spans  exceeding  500  ft.  shall  be 
avoided  if  possible. 

Railroad  Crossings. — (a)  The  span  where  line  crosses  railroad 
shall  be  kept  as  short  as  possible ;  but  hi  no  case  must  a  pole  be 
placed  a  smaller  distance  than  12  ft.  from  the  rail,  except  in  the 
case  of  sidings,  where  the  distance  may  be  reduced  to  6  ft.  At 
loading  sidings  sufficient  space  must  be  allowed  for  a  driveway 
between  rail  and  pole.  When  possible  the  distance  between  rail 
and  pole  should  not  be  less  than  the  height  of  the  pole,  but  if 
this  spacing  requires  a  span  greater  than  120  ft.,  it  will  be  prefer- 
able to  place  the  pole  nearer  to  the  rail  provided  the  ground  is 
suitable.  If  it  is  necessary  to  cross  the  railroad  with  a  span 
greater  than  150  ft.,  the  double-pole  arrangement  as  used  for 

extra  long  spans,  and  as  shown  on  drawing  No shall  be 

adopted. 

(6)  In  all  cases  the  cross-arms  and  insulators  shall  be  doubled 
on  the  poles  nearest  the  rail. 


WOOD  POLE  TRANSMISSION  LINES  381 

(c)  The  poles  at  railroad  crossings  must  be  set  not  less  than 
6  ft.  in  the  ground  (4  ft.  in  rock). 

(d)  If  the  crossing  is  at  a  spot  where  grass  or  other  fires  might 
cause  injury  to  the  poles,  these  shall  be  provided  with  a  casing 
of  concrete  at  least  2  in.  thick,  to  a  height  of  5  ft.  above  ground 
level. 

(e)  The  clearance  between  rail  and  high-tension  conductor 
shall  not  be  less  than  30  ft.,  and  the  poles  should  be  specially 
selected  for  strength  and  straightness. 

(/)  When  crossing  over  telephone  wires,  the  clearance  shall 
be  not  less  than  10  ft. 

(g)  The  poles  at  railway  crossings  must  be  securely  guyed, 
whether  or  not  there  is  a  bend  in  the  line.  If  a  departure  from 
the  straight  run  is  necessary,  special  attention  should  be  paid  to 
the  method  of  guying. 

(h)  The  poles  on  each  side  of  the  rail  shall  be  provided  with 
lightning  rods,  and  well  grounded.  Bent  iron  lightning  guards, 
as  shown  on  drawing  No shall  be  fixed  at  each  end  of  cross- 
arm  and  connected  to  the  ground  wire;  these  will  also  serve  the 
purpose  of  hook  guards,  to  engage  the  conductor  if  it  should 
become  detached  from  the  insulator.  If  the  nature  of  the  soil  is 
quite  unsuitable  for  the  purpose  of  grounding,  the  lightning 
rod  may  be  omitted;  but  if  the  pole  is  not  grounded,  two  strain 
insulators  must  be  placed  in  each  guy  wire  securing  poles  nearest 
to  rail;  the  upper  of  these  insulators  being  not  less  than  6  ft. 
distant  from  the  lowest  high-tension  conductor,  and  the  second 
insulator  being  not  less  than  8  ft.  above  ground  level. 

(i)  Special  attention  shall  be  paid  to  the  tying  of  the  con- 
ductors to  the  double  insulators  on  the  poles  at  each  side  of  the 
rail.  As  a  protection  against  damage  by  arcs  over  insulators, 
the  serving  of  No.  2  aluminum  tie  wire  shall  be  carried  far  enough 
to  ensure  that  the  conductor  is  protected  by  the  serving  or  tie 
to  a  distance  of  not  less  than  12  in.  from  the  center  of  insulator. 

( j)  In  addition  to  the  pole  number,  the  poles  on  each  side 
of  the  crossing  shall  bear  a  label  with  the  Company's  name  and 
the  voltage  (22,000  volts)  painted  thereon  in  easily  distinguish- 
able characters. 

9.  Angles  and  Curves. — Whenever  there  is  a  change  in  the 
direction  of  the  line,  a  sufficient  number  of  poles  must  be  provided 
to  prevent  the  angle  of  deviation  on  any  one  pole  exceeding  15 
degrees.  If  the  deflection  from  the  straight  run  does  not  exceed 


382 


ELECTRIC  POWER  TRANSMISSION 


5  degrees  it  is  not  necessary  to  use  a  pole  with  double  fixtures. 
When  the  deflection  exceeds  5  degrees,  poles  with  double  fixtures 
shall  be  used,  and  these  must  be  side  guyed.  When  the  "pull" 
at  corner  pole  exceeds  2  ft.  the  span  on  each  side  of  pole  shall  be 
less  than  150  ft.;  the  reduction  in  the  length  of  span  being  at  the 
rate  of  about  2}/£  ft.  per  foot  of  "pull,"  all  as  indicated  in  the 
table  accompanying  Fig.  1.  Should  it  be  necessary  to  turn  the 
line  at  a  point  where  space  is  limited,  through  an  angle  greater 


FIG.  1. 
Limit  of  length  of  spans  on  each  side  of  angle  pole  (standard  span  =  150  ft.). 


"Pull"  D 
feet 

Deflection  a 
degrees 

Short  span,  S, 
not  to  exceed: 
feet 

Remarks 

2 

2°-18' 

145 

| 

3 

3°-26' 

143 

Double  fixtures  not  nec- 

4 

4°-36' 

140 

essary. 

5 

5°-44' 

138 

6 

6°-52' 

135 

7 

8°-00' 

133 

8 
9 

9°-10' 
10°-20' 

130 

128 

Use  double   fixtures   on 

10 
11 

ll°-28' 
12°-38' 

125 
123 

poles. 
Side-guy. 

12 

13°-47' 

120 

13 

14°-56' 

118 

than  15  degrees,  two  or  more  poles  with  double  fixtures  may  be 
set  close  together,  each  pole  being  side  guyed,  or  securely  braced. 
In  all  cases  where  there  is  a  departure  from  the  straight  line,  the 
poles  must  be  set  so  that  the  cross-arms  will  bisect  the  angle. 

10.  Guying. — The  material  to  be  used  throughout  for  guys  is 
•Ke-in.  galvanized  seven-strand  steel  cable.  Where  the  wire  is 
wrapped  around  the  pole,  a  protecting  strip  of  No.  24  galvanized 
sheet  iron  shall  be  put  under  the  wire.  The  wire  shall  make  two 
complete  turns  about  the  pole. 


WOOD  POLE  TRANSMISSION  LINES  383 

The  anchoring  shall  generally  be  done  by  burying  an  anchor  log 
from  4  ft.  to  6  ft.  long,  and  bolting  thereto  a  %-in.  guy  rod. 
Other  methods  may  have  to  be  adopted  to  suit  the  varying 
nature  of  the  ground,  but  in  all  cases  it  is  important  to  ensure  a 
good  hold  and  to  see  that  the  guy  rod  is  in  line  with  the  guy  wire. 
The  angle  of  the  guy  wire  when  anchored  in  the  ground  shall  be 
approximately  45  degrees  where  circumstances  permit. 

No  strain  insulators  shall  be  used  on  guy  wires,  except  as  called 
for  at  railway  crossings;  but  all  guyed  poles  shall  be  provided 
with  lightning  rod  and  be  well  grounded.  It  is  not  intended  that 
work  be  done  on  live  wires  on  guyed  or  other  grounded  poles. 
As  a  general  rule  all  poles  shall  be  guyed  before  the  conductors 
are  strung.  Poles  must  be  guyed  at  all  points  as  mentioned 
below: 

(a)  At  angles  exceeding  5  degrees. 

(6)  Where  the  line  goes  up  a  15  per  cent,  or  steeper  grade 
(head  guys  every  fifth  or  sixth  pole,  or  only  at  top  of  hill  on 
short  lengths). 

(c)  On  hillsides  where  the  footing  may  be  good,  but  where 
there  is  danger  of  slipping  stones  or  soil  producing  side  pressures 
on  the  pole  (side  guy). 

(d)  At  each  end  of  exceptionally  long  spans,  where  double  poles 
are  used. 

(e)  All  poles  with  double  fixtures. 

11.  Insulators. — The  line  insulators  will  be  supplied  by  Messrs. 

They  will  be  of  the  pin  type,  the  pins 

having  porcelain  bases  with  wood  thimbles  and  ^-in.  galvanized 
iron  bolts  for  fixing  to  cross-arms.     The  pole-top  insulators  will 
be  supported  on  malleable-iron  pole-top  pins,  and  the  separable 
thimbles  of  these  pole-top  pins  will  be  cemented  into  the  insu- 
lators at  the  makers'  works.     The  insulators  shall  be  mounted 
on  the  cross-arms  after  the  poles  have  been  erected.     The  pole- 
top  insulator  pins  may  be  bolted  in  position  before  erection  of 
pole. 

12.  Stringing  of  Wires. — No.  2/0  seven-strand  bare  aluminum 
cable  will  be  used  throughout.     Care  must  be  used  in  handling 
the  conductors,  to  guard  against  cuts  or  scratches  or  kinks.     The 
conductor  must  not  be  drawn  over  rough  or  rocky  ground  where 
it  is  liable  to  be  injured  by  stones,  etc. 

It  is  important  that  the  cables  be  pulled  up  to  the  proper 
tension  so  that  the  sag  will  be  in  accordance  with  the  particulars 


384 


ELECTRIC  POWER  TRANSMISSION 


given  on  the  curves  Figs.  2  and  3.  These  curves  give  not  only 
the  correct  sag  at  center  of  span,  but  also  the  required  tension  in 
the  cable  at  the  time  of  stringing.  The  curves  are  calculated  for 
wires  subject  only  to  their  own  weight  and  hanging  in  still  air. 

In  the  case  of  extra  long  spans,  and  where  the  grade  is  not 
constant,  it  will  generally  be  found  more  convenient  and  quicker 
to  adjust  the  tension  by  means  of  a  spring  dynamometer  than  by 
measuring  the  sag. 

The  cables  must  not  be  pulled  around  insulator  pins  on  angle 
poles. 


100 
90 
80 

r° 

g  60 
1  50 

I40 

ft  30 

^20 

1  10 
I    ° 
-10 
-20 
-80 

\ 

\ 

V 

\ 

^ 

Tension   Temperature  Curves 

For  use  in  Stringing 
No.2/0  Stranded  Aluminium  Conductors 
(Calculated  for  Maximum  Stress=13000  Ibs.  per  sq.  in. 
with  wind  velocity  47  miles  per  hour,  combined  with 
M"ice  coating,  at-20°F.  ) 

^ 

^x 

\ 

\^ 

\^ 

X. 

\ 

X 

X 

X 

\ 

N 

X 

X. 

X 

x 

^X 

x 

X 

X 

^ 

X 

x 

X 

X 

X 

^ 

X 

X 

X 

^ 

\ 

X 

\^_ 

X 

v^ 

v 

X 

X 

^ 
^? 

^ 

**•/*•/ 

Sft 

^ 

x. 

X 

\ 

•>«.  -s 

N, 

\^ 

^ 

^ 

SJ 

1 

s 

^s 

X 

^ 

\ 

"X 

\ 

X, 

200          300'          400  500  600          700  800>          900        -10( 

"  Tension  in  Lbs.as  Indicated  .by  Dynamometer 
FIG.  2. — Chart  giving  tension  at  which  wires  should  be  strung. 


1100 


The  tie  wire  shall  be  No.  2  B.  &  S.  solid,  soft  aluminum  wire. 

The  tie  on  straight  runs  shall  be  of  the  type  known  as  the  armor 
top,  with  the  conductor  in  the  groove  on  top  of  insulator.  At 
corners,  the  tie  shall  be  of  the  type  known  as  the  armored  Western 
Union,  with  the  conductor  carried  around  the  insulator  in  the 
side  groove.  The  tie  shall  be  a  modification  of  the  type  used  by 
the  Niagara,  Lockport  and  Ontario  Power  Co.,  between  Niagara 
Falls  and  Buffalo.  The  serving  of  No.  2  tie  wire  on  the  conductor 
is  for  the  purpose  of  preserving  the  latter  from  abrasion  and  from 
damage  due  to  possible  electric  discharges  over  the  insulator. 
The  use  of  pliers  should  be  avoided  in  making  the  ties,  except  for 


WOOD  POLE  TRANSMISSION  LINES 


385 


the  final  clinching,  when  they  must  be  used  with  care  to  avoid 
cutting  or  otherwise  injuring  the  conductor. 

When  joints  are  required  in  the  conductors,  they  shall  be  made 
with  Maclntyre  tubes  which  shall  be  given  two  twists  with  the 
splicing  clamps  provided  for  the  purpose. 


100 
90 
80 

ITO 

leo 
£50 

I- 

830 

p 

§.10 

1. 

-10 
-20 
-30 

P 

/ 

«4 

4 

v> 

N 

g 

r^ 

jA 

V 

/ 

S/ 

Y 

*> 

^ 

*rf 

? 

/ 

f 

. 

/ 

/ 

' 

/ 

S 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

s 

/ 

/ 

/ 

/ 

/ 

jf 

/ 

/ 

/ 

/ 

7 

/ 

/ 

/ 

] 

/ 

/ 

/ 

/ 

/ 

/ 

1 

/ 

Sag   Temperature  Curves 
For  use  in  Stringing 
No.2/0  Stranded  Aluminium  Conductors 
(Calculated  for  Maximum  Stress=13000  Ibs.  per  sq.  in. 
with  wind  velocity  47  miles  per  hour,  combined  with 
H"ice  coating,  at-20'F.) 

/ 

1 

/ 

1 

4 

8      10121416182022242628303234363840 

Sag  at  Center  of  Span  (Inches) 
FIG.  3. — Chart  giving  sag  in  wires  when  correctly  strung. 

13.  Locating  and  Numbering  Poles. — All  poles  shall  bear  a 
distinguishing  number  in  clear  bold  figures  about  10  ft.  above 
ground  level.  These  numbers  will  correspond  with  the  numbers 
on  the  plans  which  will  be  prepared  as  soon  as  possible  after  the 
poles  have  been  erected  in  position.  '  The  plans  will  be  drawn  to  a 
scale  sufficiently  large  to  show  the  location  of  each  pole. 


25 


APPENDIX  III 

SPECIFICATIONS  FOR  STEEL  TOWER  TRANS- 
MISSION LINE 

These  specifications,  which  are  preliminary  specifications  sub- 
ject to  revision  in  minor  details  after  bids  for  the  various  materials 
have  been  received  and  considered,  cover  the  construction  of  an 
overhead  transmission  line  connecting  the  generating  station  at 

with  distributing  station  at ^ 

a  distance  of  approximately  60  miles  as  measured  along  the  right- 
of-way  of  the  transmission  line. 

Methods  of  construction  are  not  dealt  with  in  detail,  because 
the  work  hi  the  field  will  be  in  the  hands  of  a  competent  and  ex- 
perienced construction  engineer  who  will  be  allowed  considerable 
latitude  in  regard  to  the  actual  handling  of  materials  and  in  de- 
ciding upon  the  best  methods  to  be  adopted  in  the  erection  of 
towers,  stringing  of  conductors,  and  other  details  of  practical 
line  construction. 

General  Description  of  Line. — Two  three-phase  circuits  of  No. 
2/0  equivalent  copper  cable  will  be  run  in  parallel  on  one  set  of 
steel  towers  spaced  approximately  eleven  to  the  mile.  The 
towers  will  be  of  the  semi-flexible  type,  with  rigid  strain  towers 
at  intervals  of  about  a  mile,  or  more  frequently  where  corners 
or  extra  long  spans  render  then:  use  necessary.  A  J^Q-in.  gal- 
vanized Siemens-Martin  steel-strand  cable  will  be  carried  the  full 
length  of  the  line  and  be  firmly  secured  to  the  top  of  each  tower. 

The  pressure  between  conductors  will  be  80,000  volts,  and  the 
suspension  type  of  insulator  will  be  used  throughout. 

Details  of  entering  bushings  and  methods  of  connecting  light- 
ning arresters  at  generating  and  receiving  stations  are  not  dealt 
with  in  these  specifications  as  they  come  under  another  contract. 

The  proposed  line  has  been  staked  out  by  the  stadia  survey 
party,  and  the  right-of-way  secured  where  necessary.  Stakes 
have  been  driven  to  indicate  proposed  location  of  towers,  but 
these  positions  are  subject  to  modification. 

386 


STEEL  TOWER  TRANSMISSION  LINES         387 

The  line  passes  through  country  that  is  for  the  greater  part 
uncultivated;  the  ground  is  undulating  and  in  some  parts  wooded. 
A  considerable  amount  of  clearing  has  yet  to  be  done.  Roads  are 
bad;  but  the  transmission  line  is  within  2  to  3  miles  of  the  railway 
at  all  points. 

Duties  of  Engineer  in  Charge  of  Construction. — Before  the 
work  of  construction  is  begun,  the  construction  engineer  will  go 
over  the  line  as  staked  out  by  the  preliminary  survey  party  and 

as  shown  on  plan  No herewith.  He  will  take  with  him  an 

engineer  equipped  with  a  light  transit,  and  an  assistant  capable 
of  acting  as  axeman  or  rodman  as  circumstances  may  require. 
The  construction  engineer  will  decide  in  the  field  the  position  of 
each  tower,  making  changes  in  the  preliminary  plan  in  the  matter 
of  tower  locations  and  even  to  a  small  extent  in  the  route  to  be 
followed,  if  in  his  opinion  such  changes  will  result  in  a  better  and 
more  economical  line.  Hub-stakes  shall  be  driven  to  mark  the 
center-point  of  each  tower,  and  a  second  stake  shall  be  driven 
about  12  ft.  ahead  or  in  the  rear  of  the  hub-stake  in  the  direction 
of  the  line.  This  is  for  reference  when  setting  the  anchor  stubs. 

The  construction  engineer  must  check  clearances  between 
conductor  and  ground,  and  on  long  spans,  especially  if  there  is 
doubt  as  to  position  and  amount  of  minimum  clearance,  he 
should  take  the  necessary  particulars  to  allow  of  the  matter  being 
settled  in  the  office.  After  agreeing  and  checking  the  alterations 
to  plan  in  the  office,  the  construction  engineer  will  assist  in 
making  out  the  shipping  schedules  for  delivery  of  materials  at  the 
most  suitable  points. 

In  regard  to  the  work  of  erection  proper,  the  construction  en- 
gineer will  attend  to  all  details  of  organization  of  the  parties  in  the 
field,  and  will  study  the  best  means  of  distribution  of  materials 
along  the  line ;  all  with  the  view  of  avoiding  unnecessary  expendi- 
ture, and  of  carrying  out  the  work  expeditiously  and  in  a  work- 
man-like manner.  Such  details  as  the  actual  methods  to  be 
adopted  in  the  erection  of  towers  and  stringing  of  wires  will  be 
decided  upon  after  discussion  with  the  chief  engineer,  and  after 
due  weight  has  been  given  to  manufacturers'  suggestions. 

Clearing. — On  those  parts  of  the  line  on  which  clearing  is  re- 
quired, it  is  proposed  that  this  work  be  done  immediately  after 
the  line  has  been  finally  staked  out.  This  clearing  will  extend 
60  ft.  on  each  side  of  the  center  line  of  the  right-of-way,  and  it 
will  be  carried  out  under  a  separate  contract. 


388  ELECTRIC  POWER  TRANSMISSION 

Towers. — Two  standard  types  of  steel  tower  will  be  used :  these 
will  be  referred  to  as  the  strain  type  and  flexible  type  respect- 
ively. Copy  of  the  specification  on  which  bids  will  be  obtained 
from  manufacturers  is  attached  hereto. 

Foundations  for  Towers. — The  use  of  concrete  is  to  be  avoided, 
but  in  marsh  land  or  loose  soil  concrete  footings  may  be  necessary. 
The  decision  as  to  where  concrete  is  to  be  used  will  rest  largely 
with  the  engineer  in  charge  of  construction  in  the  field. 

When  the  tower  stands  on  solid  rock — which  may  occur  in  a 
few  instances — the  standard  footings  will  not  be  used;  but  a 
special  wedge  bolt,  shaped  at  the  top  to  take  the  standard  tower, 
will  be  grouted  in  with  sulphur  or  other  approved  cement.  In 
levelling  up  on  rock  foundations,  it  may  sometimes  be  cheaper 
to  build  up  one  or  two  piers  of  concrete,  securely  tied  down  to 
the  rock,  rather  than  level  off  the  rock  on  the  high  side. 

In  selecting  sites  for  towers,  the  construction  engineer  shall 
pay  attention  to  the  matter  of  foundations,  and  endeavor  to 
secure  sites  where  the  foundations  are  good.  Hillsides  are  to  be 
avoided,  especially  where  the  soil  is  liable  to  crumble  or  slide. 
The  matter  of  grading  should  also  be  considered  when  finally 
selecting  sites:  much  may  be  accomplished  in  the  judicious  selec- 
tion of  tower  sites  by  slightly  adjusting  the  length  of  span  to 
obtain  sites  which  will  tend  to  equalize  the  grade. 

To  facilitate  the  work  of  erection,  a  wooden  digging  templet 
will  be  provided,  together  with  a  rigid  but  light-weight  angle 
steel  templet  to  ensure  the  correct  placing  of  the  anchor  stubs; 
the  latter  being  bolted  to  the  templet  before  the  work  of  back- 
filling the  holes  is  commenced. 

The  second  stake  which,  as  previously  mentioned,  will  be 
driven  truly  in  line  with  the  hub  stake,  will  be  used  for  the  correct 
setting  of  these  templets.  The  steel  templet  must  be  carefully 
leveled  up  in  order  that  the  center  line  of  the  tower  shall  be 
vertical. 

Grounding. — In  cases  where  the  iron  work  of  the  foundations 
is  completely  encased  in  concrete,  the  tower  shall  be  well  grounded 
by  means  of  a  10-ft.  length  of  1-in.  galvanized-iron  pipe  driven 
or  buried  in  the  ground,  and  electrically  connected  to  one  of  the 
tower  legs.  When  the  tower  stands  on  rock,  an  effort  should  be 
made  to  obtain  a  good  ground  by  carrying  a  length  of  the  galvan- 
ized guy  wire  from  the  tower  leg  to  a  rod  driven  in  damp  soil  at  a 
short  distance  from  the  tower  if  a  suitable  spot  can  be  found. 


STEEL  TOWER  TRANSMISSION  LINES         389 

Guying. — Where  guy  wire  is  required,  the  Ke-m-  Siemens- 
Martin  steel  ground  wire  shall  be  used.  When  the  distance 
between  strain  towers  exceeds  %  mile,  one  flexible  tower  situated 
about  midway  between  the  strain  towers  shall  be  head-guyed 
in  both  directions.  Flexible  towers  used  at  corners  where 
the  deviation  lies  between  5  and  8  degrees,  and  the  approach 
spans  are  of  normal  length,  shall  be  guyed  with  two  guy  wires  so 
placed  as  to  take  the  corner  strain  and  resist  overturning  of  the 
tower  owing  to  the  resultant  pull  of  the  wires. 

Angles. — The  semi-flexible  support  is  designed  for  use  on 
straight  runs  only;  but  if  the  deviation  from  the  straight  line 
does  not  exceed  5  degrees,  these  intermediate  supports  may  be 
equipped  with  strain  insulators  and  used  at  corners.  For  angles 
greater  than  5  degrees,  but  not  exceeding  a  limit  of  8  degrees,  these 
supports  may  be  used  with  two  guy  wires  to  take  the  transverse 
stress  due  to  the  resultant  pull  of  the  wires,  if  the  approach 
spans  are  reduced  to  240  ft.,  an  8-degree  curve  may  be  turned 
on  a  semi-flexible  structure  without  guy  wires. 

Strain  towers  shall  be  used  for  turning  corners  up  to  30  degrees ; 
but  when  the  total  deviation  exceeds  this  amount,  two  towers 
must  be  used. 

Erection  of  Towers. — The  actual  organization  of  the  various 
crews  for  distributing  material,  setting  anchor  legs,  assembling 
and  erecting  the  towers,  will  be  left  to  the  engineer  in  charge  of 
construction,  who  will  so  conduct  operations  as  to  carry  out  the 
work  efficiently  at  the  lowest  possible  cost.1 

Insulators. — These  will  be  of  the  suspension  type  to  comply 
with  the  requirements  detailed  in  the  insulator  specification  of 
which  copy  is  attached  hereto. 

Conductors. — The  conductors  shall  be  19  strand  hard-drawn 
copper  cables  equivalent  in  section  to  00  B.  &  S.  gauge.  The 
tensile  strength  of  the  finished  cable  shall  not  be  less  than  90  per 
cent,  of  the  strength  of  the  individual  wires  forming  the  cable; 
and  these  shall  satisfy  the  strength  requirements  of  the  standard 
specification  drawn  up  by  the  American  Society  for  Testing 
Materials. 

1  Mr.  R.  A.  Lundquist's  book  on  "Transmission  Line  Construction"  is 
of  special  interest  to  the  engineer  in  charge  of  construction.  Excellent 
articles  describing  practical  methods  of  construction  also  appear  from  time 
to  time  in  the  technical  press.  The  article  by  Mr.  A.  B.  Cudebec  on  "Steel 
Tower  Transmission  Line  Construction"  in  the  Electrical  World  of  July  17, 
1915,  contains  much  valuable  information. 


390  ELECTRIC  POWER  TRANSMISSION 

The  electrical  conductivity  shall  not  be  less  than  97  per  cent, 
by  Matthiessen's  standard. 

The  total  weight  of  copper  conductor  required  is  estimated  at 
800,000  Ib.  It  shall  be  delivered  on  drums  or  reels  each  contain- 
ing 1  mile  of  cable. 

Joints  in  Conductors. — The  splices  shall  be  made  with  copper 
sleeves  of  the  "Maclntyre"  or  similar  approved  type.  The 
finished  joint  shall  consist  of  3  turns.  The  tools  provided  for  the 
purpose  shall  be  used  in  making  the  joints. 

Spans  and  Wire  Stringing. — The  actual  method  of  stringing 
the  wires  will  be  left  to  the  judgment  and  experience  of  the  con- 
struction engineer.  It  is,  however,  suggested  that  three  conduc- 
tors be  drawn  up  at  a  time,  using  the  arrangement  of  sheave 
blocks  known  as  an  "equalizer." 

The  average  span  shall  be  approximately  480  ft.  This  may  be 
increased  to  a  limit  of  500  ft.  between  flexible  supports,  and  to  a 
limit  of  1200  ft.  between  two  strain  towers  without  intermediate 
supports.  It  is  thought  that  three  or  four  points  on  the  line  may 
advantageously  be  spanned  between  two  strain  towers  placed 
from  1000  to  1200  ft.  apart.  In  the  case  of  abnormally  long 
spans,  it  is  important  to  see  that  the  contour  of  the  ground  is  such 
as  to  allow  of  maximum  sag  while  maintaining  the  specified  mini- 
mum clearance  between  H.  T.  conductors  and  ground. 

The  clearance  between  lowest  wire  and  ground  shall  in  no  case 
be  less  than  28  ft. 

The  charts  Nos and give  all  necessary  particu- 
lars for  the  stringing  of  conductors  and  guard  wire  at  various 
temperatures.  The  guard  wire  connecting  the  tops  of  all  towers 
shall  be  strung  and  securely  clamped  to  the  steel  structure  before 
the  conductors  are  drawn  up.  Dynamometers  will  be  provided, 
and  their  use  is  recommended,  especially  when  spans  are  unequal 
in  length,  and  on  extra  long  spans  between  two  strain  towers. 
If  an  equalizer  is  used,  it  is  not  necessary  to  insert  a  dynamometer 
in  more  than  one  leg.  Special  attention  shall  be  paid  to  the 
drawing  up  of  cables  to  the  proper  tension  or  sag.  Too  great  a 
sag  is  almost  as  objectionable  as  too  great  a  tension;  but  it  must 
be  remembered  that  where  a  dip  occurs  in  the  line  of  supports 
there  is  sometimes  a  possibility,  in  very  cold  weather,  of  the 
conductor  being  drawn  up  (by  contraction)  above  the  proper 
level  of  the  lowest  insulator.  The  construction  engineer  should 
watch  for  this  possibility  with  a  view  to  guarding  against  it. 


STEEL  TOWER  TRANSMISSION  LINES         391 

When  the  conductors  have  been  drawn  up  and  transferred 
from  snatch  block  to  insulator  clamp,  it  is  important  to  see  that 
the  suspension  insulator  hangs  truly  vertical  before  finally  tight- 
ening up  the  clamp. 

There  will  be  no  transpositions  on  the  H.  T.  conductors.  The 
telephone  wires  are  run  on  a  separate  set  of  wood  poles  and  they 
will  be  transposed  at  every  support. 

SPECIFICATION  FOR  STEEL  TOWERS 

These  towers  are  for  use  on  an  80,000-volt,  three-phase  trans- 
mission line  using  insulators  of  the  suspension  type.  It  is 
proposed  to  use  two  standard  types  of  towers  only;  these  will  be 
referred  to  as  the  rigid  or  strain  towers,  and  the  flexible  towers. 

The  strain  towers  shall  be  designed  with  four  corner  legs  and 

square  bases,  generally  as  indicated  on  plan  No herewith. 

An  effort  will  be  made  to  avoid  the  use  of  special  structures,  and 
where  extra  long  spans  have  to  be  carried,  two  standard  strain 
towers  may  be  placed  close  together.  In  one  or  two  places  it 
may  be  necessary  to  use  extra  high  towers,  and  it  is  proposed  to 
use  the  standard  tower  mounted'  on  a  special  base,  generally 

as  shown  on  plan  No ,  designed  to  raise  the  tower  18  ft., 

or  such  other  amount  approximating  to  this  dimension  as  may 
best  suit  manufacturers'  designs. 

The  intermediate  or  flexible  type  of  support  will  be  of  the  "  A  "- 

frame  design,  generally  as  shown  on  plan  No Preference 

will  be  given  to  a  design  consisting  of  few  parts,  provided  this  will 
not  add  appreciably  to  the  cost  of  transporting  the  towers  over 
rough  roads  to  the  point  of  erection.  The  parts  of  all  towers 
shall  be  galvanized  when  ready  for  assembling;  but,  in  the  case 
of  the  flexible  type  of  structure,  an  alternative  offer  for  painted 
steel  work  will  be  considered,  provided  the  number  of  parts  is 
small  and  the  section  of  metal  reasonably  large. 

The  plans  referred  to,  which  accompany  this  specification,  give 
all  necessary  leading  dimensions;  but  the  cross-section  of  the  vari- 
ous members  and  the  details  of  design  are  left  to  the  manufac- 
turer, who  is  also  at  liberty  to  submit  alternative  proposals.  In 
no  case  must  the  distance  between  conductors  be  less  than  8^ 
ft.  or  the  height  above  ground  of  the  point  of  attachment  of 
insulators  on  lowest  cross-arm  less  than  40  ft.  The  sections  of 
structural  steel  used  for  the  main  corner  members  of  the  strain 


392  ELECTRIC  POWER  TRANSMISSION 

towers  or  for  the  main  members  of  the  flexible  towers  shall  not 
be  less  than  ^  in.  thick,  and  no  metal  less  than  %g  in.  thick 
shall  be  used  in  the  construction  of  these  towers. 

Number  of  Towers  Required. — Offers  shall  be  based  on  the 
following  quantities,  which  are  subject  to  slight  modification. 

Flexible  towers  (plan  No ) 592 

Rigid  towers  (plan  No ) 65 

Extension  bases  (plan  No ) 4 

Working  and  Test  Loads  for  Towers. — The  normal  length  of 
span  is  480  ft.  and  the  total  vertical  load  per  tower,  consisting  of 
six  conductors  and  one  guard  wire  together  with  estimated 
possible  ice  loading  and  the  weight  of  the  six  insulators,  is  3100 
lb.;  but  the  spans  will  in  many  cases  exceed  the  aver  age  length. 
The  maximum  total  overturning  pressure  in  a  direction  at  right 
angles  to  the  line,  due  to  wind  blowing  across  the  wires,  is  esti- 
mated at  3300  lb. ;  this  may  be  considered  as  distributed  equally 
between  the  points  of  attachment  of  the  seven  wires.  The  manu- 
facturer should  estimate  the  pressure  of  wind  on  the  tower 
structure  itself  by  allowing  a  maximum  pressure  of  13  lb.  per 
square  foot  of  tower  surface.  .  A  factor  of  safety  of  2^  shall  be 
used  in  making  stress  calculations. 

One  tower  of  each  type  shall  be  tested  in  the  presence  of  pur- 
chaser's representative,  and  must  withstand  without  exceeding 
the  elastic  limit  of  the  steel,  or  suffering  appreciable  permanent 
deformation,  the  following  test  loads  applied  at  the  points  indi- 
cated on  the  plans  previously  referred  to.  These  tests  are  to 
be  made  with  the  tower  erected  on  its  own  foundations  in  such 
a  manner  as  to  reproduce  as  nearly  as  possible  the  conditions 
under  which  it  will  ultimately  be  erected. 

Strain  Tower  Test  Loads.— (1)  A  breast  pull  of  15,000  lb. 
applied  in  the  direction  of  the  line  at  the  point  of  attachment  of 
the  middle  cross-arm. 

(2)  A  vertical  load  of  1000  lb.  applied  at  the  end  of  any  cross- 
arm. 

(3)  A  torsional  load  of  3500  lb.  applied  in  a  direction  parallel 
to  the  line  at  the  end  of  any  cross-arm. 

Flexible  Tower  Test  Loads.— (1)  A  transverse  pull  of  4500  lb. 
applied  in  a  direction  at  right  angles  to  the  line  at  the  point  of 
attachment  of  the  middle  cross-arm. 

(2)  A  vertical  load  of  800  lb.  applied  at  the  end  of  any  cross- 
arm. 


STEEL  TOWER  TRANSMISSION  LINES         393 

Metal  steps  shall  be  provided  on  all  towers  within  8  ft.  of  ground 
level  for  the  use  of  linemen. 

It  is  requested  that  manufacturers  tendering  for  steel  towers 
call  attention  to  any  features  of  the  particular  design  proposed 
which  may  tend  to  reduce  cost  of  transport  and  erection  on  site, 
as  these  are  matters  which  will  receive  consideration  when  placing 
the  contract. 

Galvanizing  Test. — The  purchaser  reserves  the  right  to  reject 
all  towers  of  which  the  galvanizing  is  not  of  the  best  quality. 
Tests  will  be  made  before  erection  as  follows: 

Samples  of  steel  work  will  be  immersed  in  a  solution  of  sul- 
phate of  copper  (specific  gravity  about  1.185)  maintained  at  a 
temperature  of  60  to  70°  F.  After  remaining  in  the  solution  1 
minute,  the  sample  will  be  removed,  thoroughly  washed  in  water, 
and  wiped  dry.  This  process  will  be  repeated  four  times,  after 
which  there  must  be  no  appearance  of  red  spots  indicating  copper 
deposit. 

SPECIFICATION  FOR  PORCELAIN  LINE  INSULATORS 

Number  of  Insulators  Required. — The  approximate  quantities 
required,  as  based  on  preliminary  estimates  are: 

Suspension  type 4000 

Strain  type 880 

Climatic  Conditions. — The  transmission  line  on  which  the  insu- 
lators will  be  used  is  located  in  the district,  where 

severe  thunder  storms  and  heavy  rain  may  be  expected  during 
the  summer  months,  and  where  sleet  storms  and  low  tempera- 
tures are  prevalent  in  the  winter. 

Working  Voltage. — The  transmission  is  three-phase  off  delta 
connected  transformers,  at  a  frequency  of  60  and  a  maximum 
working  pressure  of  84,000  volts  between  wires. 

Design  of  Insulators. — The  design  of  the  suspension  type  and 
strain  insulators  is  left  to  the  manufacturer,  who  must  submit 
dimensioned  drawings  or  samples  with  his  offer.  The  units 
making  up  the  strain  insulators  need  not  necessarily  differ  in  de- 
sign from  the  units  of  the  suspension  insulators,  provided  the 
latter  are  capable  of  withstanding  the  mechanical  tests  required 
for  the  strain  insulators.  The  towers  have  been  designed  on  the 
assumption  that  the  weight  of  one  complete  string  of  unit  insula- 


394  ELECTRIC  POWER  TRANSMISSION 

tors  will  not  exceed  60  Ib.  and  that  the  distance  between  point  of 
suspension  and  conductor  will  not  exceed  36  in.  These  limits 
should  not  be  exceeded.  It  is  preferred  that  the  number  of 
units  in  the  complete  string  be  not  less  than  three  nor  more  than 
five. 

Metal  Parts. — All  metal  parts  subject  to  rust  and  corrosion, 
such  as  malleable  iron  castings  and  steel  forgings,  shall  be  heavity 
galvanized  and  capable  of  withstanding  the  usual  tests. 

Glaze. — The  surfaces  of  the  porcelain  not  in  contact  with  the 
cement  shall  be  uniformly  coated  with  a  brown  glaze,  free  from 
grit. 

Cement. — Pure  Portland  cement  only  shall  be  used  in  as- 
sembling the  parts  of  the  unit  insulator. 

Mechanical  Tests. — An  inspection  will  be  made  of  all  insulators 
with  the  object  of  rejecting  those  containing  open  cracks  in  glaze 
or  porcelain. 

One  complete  suspension  insulator,  selected  at  random,  and 
consisting  of  the  requisite  number  of  units,  shall  withstand  a  load 
of  5000  Ib.  without  rupture  or  sign  of  yielding  in  any  part. 

At  least  three  units  of  which  the  strain  insulators  are  built  up 
shall  be  tested  to  the  breaking  limit,  and  must  withstand  an  ulti- 
mate load  of  not  less  than  12,000  Ib. 

Electrical  Tests. — Three  or  four  complete  insulator  strings, 
both  suspension  type  and  strain  type,  shall  withstand  without 
flash  over  a  "  wet "  test  of  200,000  volts.  In  all  cases  the  electrical 
stress  shall  be  applied  for  1  minute,  and  the  spray  shall  be  directed 
upon  the  insulator  at  an  angle  of  45  degrees  under  a  pressure 
of  40  Ib.  per  square  inch  at  the  nozzles,  the  precipitation  being 
at  the  rate  of  1  in.  in  5  minutes.  The  suspension  insulators 
shall  be  hung  vertically,  and  the  strain  insulators  horizontally. 

The  connection  of  the  test  wires  shall  be  so  made  as  to  re- 
produce as  nearly  as  possible  the  working  conditions. 

The  manufacturer  shall  satisfy  himself  by  his  standard  factory 
tests  that  each  unit  is  sound  mechanically  and  electrically.  The 
"dry"  flashover  of  the  complete  string  of  insulator  units  shall 
not  be  less  than  240,000  volts;  but  this  test  need  not  be  made  in 
the  presence  of  the  purchaser's  representative. 

The  transformer  used  for  the  electrical  tests  must  be  capable  of 
a  reasonably  large  k.v.a.  output;  the  e.m.f.  wave  shall  be  as  nearly 
as  possible  sinusoidal,  and  the  frequency  shall  be  within  the 
limits  of  25  and  60  cycles. 


STEEL  TOWER  TRANSMISSION  LINES         395 

Packing  of  Insulators. — It  is  desirable  that  the  parts  for  one 
complete  insulator,  or  at  most  for  two  insulators,  be  packed 
complete  in  a  separate  barrel  or  crate,  and  that  the  contents 
be  clearly  described  on  attached  label. 

Wire  Clamps. — A  suggested  clamp  for  use  with  suspension 

insulators  is  shown  on  drawing  No herewith;  and  drawing 

No shows  a  proposed  strain  insulator  clamp.  Makers  are 

asked  to  submit  samples  or  drawings  of  their  standard  types, 
preferably  of  the  general  design  indicated  by  the  above-mentioned 
drawings.  The  conductor  to  be  carried  is  an  equivalent  No.  2-0 
gauge  (B.  &  S.)  stranded  copper  cable;  the  groove  for  the  wire 
should  be  slightly  curved  and  flared  at  the  ends. 


INDEX 


Aluminum  cell  arresters,  187 
conductors.     (See  Wire.) 

compared   with   other  mate- 
rials, 48, 69 

Anchors  for  guy  wires,  383 
Angles  and  curves  on  transmission 

lines,  381,  389 
Arcing-ground  suppressor,  194 

rings  and  horns,  177 
Arresters.     (See    Lightning    arrest- 
ers.) 
Asphalt  troughing  for  underground 

cables,  208 
Auxiliary  steam-driven  plant,  43 

B 

Bi-metallic  conductors,  69,  72 
"Boosters"  for  voltage  control,  113 
Brush  treatment  of  pole  butts,  311 
Bushings,  condenser  type,  150 
entering,  143 
insulating,  design  of,  146 
Butts,  pole,  preservative  treatment 

of,  309 
reinforcing  decayed,  312 


Cables.     (Refer  also  to  Wire.) 

overhead,    stranded    compared 

with  solid  wires,  263 
with  hemp  core,  73 
with  steel  core,  73 
underground,  6,  9,  38,  199 
advantages  for  D.  C.  trans- 
mission, 244,  200 
as  protection  against  surges, 

191 
capacity  of,  98,  212,  217 


Cables,   underground,    construction 

of,  202 

cost  of,  199,  208 
design  of,  212,  221 
grading,  216 
joints  in,  228 
losses  in,  223 
methods  of  laying,  206 
reactance  of,  214,  217 
submarine,  200 
temperature  rise  of,  225 
terminals,  211 
voltage  limitations,  201 
Capacity,  130 

current,  29,  31,  99,  217,  223 

line  losses  due  to,  112 
distributed,  101 
effect  of,  on  regulation,  28 
in  line  insulators,  135 
in  terms  of  inductance,  100 
of  suspension  insulators,  135 
of  three-phase  lines,  30,  32,  97 
of  transmission  lines,  28,  96 
of  underground  cables,  98,  212, 

217 

pressure  rise  due  to,  29,  109 
Catenary  curve  for  sag  calculations, 

249 
Charging    current.     (See    Capacity 

current.) 
Choke  coils,  192 
Clearance  between  conductors  and 

pole,  158 
between     entering     wires     and 

building,  144 

between  overhead  wires,  157 
Clearing  ground,  375,  387 
Coefficient  of  linear  expansion,  75 

of  self-induction,  88 
Compression    members    of    towers, 

stresses  in,  337 
wood  pole  in,  323 


397 


INDEX 


Concrete  for  tower  foundations,  344 

poles.     (See  Poles,  concrete.) 
Condensers  for  lightning  protection, 

189 

Condenser  type  of  bushing,  150 
Conductivity  of  conductor  materials, 

75 
Conductors.     (See  Cables:  Wires.) 

physical  constants  of,  74 
Conduits   for   underground    cables, 

207 
Constant    current    system.     (See 

Thury  system.) 
pressure  transmission,  119 
Continuity  of  service,  3,  7,  42 
Continuous    currents,    transmission 

by.     (See  Thury  system.) 
Copper-clad  wire,  69,  72 

conductors,  70 
Corona,  151 

as  "safety  valve,"  156 
losses  due  to,  152,  154 
voltage  formulas,  154 
Costs.     (Refer  to  subject  or  item.) 
Creosote  oil  for  wood  preservation, 

311 

Critical   temperature    (sag    calcula- 
tions), 284,  286 

voltage  of  corona  formation,  153 
Cross  arms,  139,  377 
Current    density    in    underground 

cables,  225 
total,  in  line  with  appreciable 

capacity,  110 

Curves  and  angles  on  transmission 
lines,  381,  389 


Deflection.     (See  Sag;  Poles;  Tow- 
ers.) 

Depreciation,  50,  53,  55,  59,  61 
Dielectric    circuit     (fundamentals), 

130 

constant,  99,  130,  213 
flux,  130 

Direct   current  transmission.     (See 
Thury  system.) 


Distance  of  transmission  (economic 
limits),  36 

Disturbances  due  to  switching  op- 
erations, 173 

Ducts  for  underground  cables,  207 

Duplex  wire  (copper-clad  steel),  69, 
72 

Duplication  of  transmission  linefc,  3, 
10,  42,  368,  373 


E 


Earthing.     (See  Grounding.) 
Economic  conductor  section,  48,  55 
considerations,  general,  2,  5,  7, 

10,  12,  13,  36,  60 
ohmic  pressure  drop,  52,  65 
voltage,  7,  53,  59,  66 
Elastance,  147 

Elastic  limit  (various  materials),  75 
modulus,  75 

transmission  line.     (See   Flexi- 
ble.) 

Electrolysis,  231 
Electrostatic  induction,  122 
Energy  stored  in  magnetic  and  elec- 
tric fields,  164 
Erection  of  poles  and  towers,  378, 

359,  389 
Erection     of     wires.     (See     Wires, 

stringing  of.) 

Estimates    for    complete    overhead 
lines,  43 


Factors  of  safety.     (See  Safety  fac- 
tors.) 

Farad,  130 

Faults   on   overhead   transmissions, 

125,  162 
in  underground  cables,  228 

Flash-over     distances     (insulators), 
133 

Flash-over  voltage  (suspension  insu- 
lators), 138,  140 

Flexible  steel  towers,  37,  331,  343, 

349,  392 
tower  lines,  331 


INDEX 


399 


Flux,  dielectric,  130 
Foundations.     (See  Pole;  Tower.) 
Frequency  in  relation  to  telephone 
troubles,  125 


Gains,  377 

Galvanizing,  test  for,  393 

versus  painting  for  steel  towers, 

41 
Garton-Daniels    lightning    arrester, 

186 

Glass  insulators,  127 
Grade,  lines  carried  up  steep,  261 
"Grading"  horn  lightning  arresters, 

183 
Grading  transmission  lines,  377 

underground    insulated    cables, 

216 

Graphical  statics  applied  to  sag-ten- 
sion calculations,  249 
Grounding,   methods  of,    178,   379, 

388 

neutral    of    three-phase    trans- 
mission, 23,  120 
Ground  resistance,  238 
Guard  rings  on  insulators,  177 

wires,  176,  196 

Guying  steel  towers,  348 

wood  poles,  382 


Inductance  of  three-phase  lines,  33, 

363 

with  any  arrangement  of  con- 
ductors, 34,  361 
Induction,    electrostatic    (telephone 

interference),  122 
magnetic,  123.      (Refer  also  to 

Inductance;  Reactance.) 
Insulation,    comparative,    of   A.    C. 
and    D.    C.    transmissions, 
237 

Insulator  materials,  127 
Insulators,  cost  of,  42 

design  of,  129,  132,  142,  393 
deterioration  of,  162 
"electrose,"  127 
factors  of  safety,  159 
flash-over  voltage,  160 
glass,  127 
pin  type,  131 
porcelain,  127 
rating  of,  161 
suspension  type,  134 
testing,  302 
ties  for,  302 
weight  of,  162 

Interruptions  to  service,  3,  7,  42 
"  Intersheath "  in  insulated  cables, 

203 

Iron  (or  steel)  for  overhead  conduc- 
tors, 69,  71,  83,  300 


H 


Hemp  core  cables,  73 

Horn  gap  lightning  arresters,  180 

Howard  asphalt  troughing,  208 


Ice  and  snow,  effects  of,  on  wires,  5, 

265,  275 

Impedance  of  power  lines.     (See  Re- 
actance, Inductance.) 
"natural,"  165,  173 
Inadequacy,  61 
Inductance  of  power  lines,   14,  24, 

79,  84,  88,  361 
in  terms  of  capacity,  100 


Joints  in  cables,  228 

in   overhead    conductors,    301, 
390 

Junction  between  overhead  and  un- 
derground conductors,  211 


K 


Kelvin's  law,  36,  49,  57,  66 


Leads,  entering,  143 
Life  of  poles  and  towers.     (Refer  to 
subject  or  item.) 


400 


INDEX 


Lightning  arresters,  175 
aluminum  cell,  187 
condenser  type,  189 
Garton-Daniels,  186 
horn  gap,  180 
low-equivalent,  183 
multi-gap,  183 
separation  of,  192 
water  jet,  179 
protection    of    overhead    lines 

from,  163,  174,  195 
rod,  175 

Line  drop,  53,  104,  108,  363 
Loading  of  wires,  usual  assumptions, 

270 
Losses  in  transmission  lines,  13,  20, 

22,  56,  57,  64,  110 
cost  of,  50,  53,  57,  64,  66 
underground  cables,  223 

M 

Magnetic  induction  (telephone  inter- 
ference), 123.  (Refer  also 
to  Inductance;  Reactance.) 

Materials  for  conductors,  48,  69 

Mershon  diagram,  90 

Modulus  of  elasticity,  75,  279 

Mosciki  condensers,  190 

N 

Natural  frequency,  167 
impedance  of  line,  165 


Parabola   and    catenary   compared, 

248 

Patrolling  transmission  lines,  9 
Permittance,  130 
Physical    constants    of     conductor 

materials,  74 
Pin  type  insulators,  131 
Pole  butts,  preservative  treatment 

of,  309 

reinforcing  decayed,  312 
foundations,  320 


Pole  lines,  wood,  for  high  pressures, 

4,  48,  303 
typical,  4,  306 
specifications  for,  375 
Poles,  concrete,  303,  324 
cost  of,  39,  325 
factors  of  safety,  329 
life  of,  39,  324 

strength  and  stiffness  of,  327 
weight  of,  325 
corner,  load  carried  by,  322 
steel,  4,  37,  301,  303,  329,  334 
wood,  300,  303,  305,  376 

"A"  and  "H"  type,  37,  303, 

307 

cost  of,  39,  41,  48 
deflection  (stiffness)  of,  318 
depth  of  holes  for,  321 
factors  of  safety,  315 
guying,  382 

insulating  qualities  of,  313 
life  of,  39,  304,  307 
preservative     treatment     of, 

309 

setting  (erecting),  378 
spacing  of,  3,  37,  321 
strength  of,  314 
weight  of,  313 

Potential  gradient,  131, 147,  149,  152 
"Pot-heads,"  211 
Power  factor,  19,  21,  23,  104,  120 
control  of,  115,  117 
of  several  circuits  in  parallel, 

120 

of  three-phase  circuit,  34 
of  underground  cables,  224 
losses.     (See  Losses.) 
maximum  on  a  single  transmis- 
sion line,  10 
stations,  cost  of,  60 
total,  transmitted  by  polyphase 

system,  21 

by  three-phase  line,  33 
Preliminary    work;    planning    new 

lines,  5,  387 
Preservative  treatment  of  pole  butts, 

309 

Pressure   available  at  intermediate 
points  on  line,  94 


INDEX 


401 


Pressure,  barometric,  at  various  alti- 
tudes, 155 
control,  112,  116 
drop,  53,  104,  108,  363 
limits  on  overhead  lines,  158 

on  underground  cables,  201 
rise  due  to  capacity,  29,  109 
rises  on  lines  r  arrying  large  cur- 
rents, 165,  197 

surges  due  to  switching  opera- 
tions, 165,  173 
Props  or  struts  (wood),  323 
Protection    of    lines    against    over- 
voltages.     (See  Lightning.) 


Quarter  wave  length  line,  163,  169 


R 


Radius  of  gyration  (tower  design), 

338 

Railroad  crossings,  9,  271 
Reactance,  72,  83,  85,  87.     (Refer 

also  to  Inductance.) 
of  underground  cables,  214 
Reactors,  rotary,  to  control  voltage, 

117 
Regulation,  voltage,  24,  56,  89,  108 

effect  of  capacity  on,  28 
Reserve  steam  driven  plant  at  re- 
ceiving end  of  line,  43 
Resistance   in   ground   connections, 

23,  181 

insulation,  of  cables,  213 
losses  due  to,  14 
necessary    to    prevent    oscilla- 
tions, 166 
of  earth  as  return  conductor, 

238 
table  (conductor  materials),  75, 

76,  85 
to    alternating    currents.     (See 

Skin  effect.) 

Resistivity  of  cable  insulation,  214 
Resonance,  167 
Roof  outlets,  143 


S 


Safety  factors,  line  insulators,  159 
poles,  315,  329 
towers,  392 
wires,  7,  301 

Sag,    aluminum    and    copper    com- 
pared, 298 
calculation  of,  251 
•  effect  of  wind,  ice,  and  tempera- 
ture on,  281 

with  supports  at  different  eleva- 
tions, 257,  260,  290 
Self-induction.     (See  Inductance.) 
Separation  between  overhead  wires, 

157 

Series  system.     (See  Thury  system.) 
Service,  continuity  of,  3,  7,  42,  125 
Single-phase  transmission,  14,  369 
Skin  effect,  15,  71,  77,  84,  85 
Sleet.     (See  Ice.) 

Snow  and  ice,  effects  of.     (See  Ice.) 
Spacing  of  overhead  wires,  156 
of  lightning  arresters,  192 
Span,  influence  of,  on  cost,  39 

length  of,  3,  37,  38,  298,  304, 

380,  390 

Spans  on  steep  grades,  256,  294 
Spark-gap  lightning  arresters,  180 
Sparking  distances,  132,  160,  181 
Specification  for  steel  tower  line,  386 

for  wood  pole  line,  375 
Specific  inductive  capacity,  99,  130, 

213 
resistance   of   cable   insulation, 

214 

Standing  waves,  173 
Steel  cored  cables,  73 

masts  or  poles,  329,  330 

towers.     (See  Towers.) 

wires   and   cables.     (See   Wire, 

iron.) 

Steep  grades,  lines  on,  261 
Stresses  in  wires  due  to  wind  and  ice, 

264 

"String  efficiency"  of  insulators,  137 
Stringing  wires.     (See  Wires.) 
Struts  or  props,  wood,  strength  of, 
323 

UMMfV 

<rrAT*  TBACHt 

HAMTA  BARBARA.  CAUPOftNIA 


402 


INDEX 


Substations,  outdoor,  10,  11 
Supports  at  different  levels,  257,  260, 

290 
comparison  of  various  types,  4, 

37,  39,  47,  300,  303 
determining  position  of,  on  un- 
even ground,  348 
flexible.     (See  Towers.) 
Surge  impedance,  165,  173 
Surges,  pressure,  165,  173,  182 
Suspension  type  insulators,  134 
Swinging  of  wires  in  high  wind,  273 
Switchgear,  simplicity  desirable,  11 
Switching  operations,  effects  of,  173 
Symbols,  list  of,  xvi 
Synchronous  machinery  to  control 
voltage,  116 


Tank  treatment  of  pole  butts,  311 
Telephone  circuit,  9,  122 
Temperature,  "critical"  (sag  calcu- 
lations), 284 
effect  of,  on  wires,  261,  278,  281, 

289 

rise  of  underground  cables,  225 
Tension  in  wires,  250,  281,  288 
Three-phase  transmission,  17,  32 
Thury  system  of  D.  C.  transmission, 

233 
compared    with   three-phase, 

238 

losses  in,  14,  238,  240 
switchgear  required  with,  245 
voltage  limits,  242,  244 
with  ground  return,  238,  242 
Ties  on  insulators,  381 
Tower  foundations,  344,  388 

cost  of,  41 
lines,  limiting  angles  on,  389 

typical  actual,  10 
Towers,   steel,    37,    329.     (See   also 

Poles,  steel.) 
cost  of,  39,  40 
dead-ending,  on  flexible  lines, 

332 

design  of,  333,  336 
erection  of,  359,  389 


Towers,  steel,  factors  of  safety  for, 

392 
flexible  type  of,  37,  331,  343, 

349,  392 

guying  "rigid"  type  of,  348 
height  of,  39 
life  of,  347 
loads  to  be  resisted  by,  330, 

335,  392 
spacing    of    (spans),    4,    299, 

330,  331,  335 
specification  for,  391 
stiffness  (deflection)  of,  344, 

349 

strength  of,  337 
thickness  of  metal  in,  347 
typical  designs  of,  330 
weight  of,  40 
wood,  303 

Transmission  lines,  cost  of,  43 
typical  existing,  4,  10 
Transposing  wires,  35,  122,  124,  367 
Travelling  waves,  170,  190,  196 
Tubes  for  tower  members,  338 
Two-phase  transmission,  15 


U 


Underground   cables.     (See   Cables, 

underground.) 

and    overhead    systems    com- 
pared, 1,  6 


Voltage.     (See  also  Pressure.) 
control,  112 
critical,  153 
drop,  52,  104,  108,  363 
economic,  7,  52,  59 
gradient,  131,  147,  149,  152 
grading  of  underground  cables, 

209 

high,  on  existing  lines,  10,  158 
line,  formula  for  estimating,  54 
practical  limitations  of,  158 
regulation,  24,  56,  89,  108 
rise  due  to  capacity,  29,  109 


INDEX 


403 


w 


Wall  outlets,  143 
Wasted  energy.     (See  Losses.) 
Water  jet  arresters,  179 
Wave  impedance,  165 

length,  168 

Waves,  electric,  170,  173,  190,  196 
Weight  of  conductors,   relative,   in 

different  systems,  20 
Weights.     (Refer  to  sub j  ect  or  item . ) 
Wind  pressure  on  poles  and  towers, 

5,268 

on  wires,  5,  267,  274 
swinging  of  wires  in,  273 
velocities    and    pressures,    267, 

336 
Wire,  aluminum,  69,  70,  298 

with  steel  core,  69 
copper,  70 

copper-clad  steel,  69,  72 
guy,  322 

iron  (and  steel),  69,  71,  83,  301 
length  of,  as  affected  by  temper- 
ature and  tension,  261,  278 
in  span,  254,  259 
materials,     physical     constants 

of,  75,  279 
table,  76 
tie,  302 
zinc,  83 
Wires,  cost  of,  46,  48 

relative  on  different  systems, 

19,  238 

deflection  of.     (See  Sag.) 
different  sizes  of,  in  same  span, 

288 

economic  size  of,  48,  55.     (Look 
also  under  Economic.) 


Wires,  effect  of  temperature  on,  261, 

278,  281,  289 

of  wind  and  ice  on,  5,  265, 
267,  274 

erection  of.     (See  Wires,  string- 
ing.) 

factors  of  safety,  7,  301 

guard,  176,  333 

joints  in,  301 

loading  of  (usual  assumptions), 
270 

reactance  of,  internal,  82,  85 

resistance  of,  75,  76,  84 

sag  of.     (See  Sag.) 

scrap  value  of,  63 

separation  of  (spacing),  156 

stringing,  46,  140,  383,  390 

swaying  of,  in  wind,  273 

telephone,  9,  122 

tension  in,  250,  281,  288 

transposition  of,  124,  367 

weight  of,  75,  76,  85 

relative,  for  different  systems, 

20 
Wood  poles.     (See  Poles.) 

pole  lines  for  high  voltages,  4, 

48,  303 
typical,  306 


Young's  modulus,  75,  279 


Zinc  as  a  material  for  overhead  con- 
ductors, 83 


SUPPLEMENTARY  INDEX 

TO  FORMULAS,   CURVES,   AND  TABLES 

For  use  in  connection  with  practical  transmission  line  calculations 
Arc-over  voltages  (insulators) 133,  134,  138,  140,  141 

Barometric  pressure  at  different  elevations 155 

Bushings,  entering,  design  of  (formulas) 147 

(example) 148 

Cables,  underground,  design  of  (example) 221 

Capacity  current  (formulas) 29,  99,  100,  110,  112,  218,  219 

(examples) 31,  105,  107 

of  overhead  lines  (formulas) 96 

of  underground  cables  (formulas) 213,  218 

(example) 221 

Clearance  between  wires 157 

for  entering  wires 144 

Concrete  poles,  weight  of 325 

Conductors.     (See  Cables,  Wires.) 

Core  diameter,  economical  (insulated  cables) 215 

Corona  (formulas) 154,  155 

Cost  of  complete  transmission  lines 45,  47 

of  hydro-electric  power  stations 60 

of  insulators 43 

of  underground  cables 210 

transmission  lines 209 

of  wood  poles 41 

"Critical"  temperature  (chart) 286 

Depreciation  (table) 62 

Dielectric  constants  of  insulating  materials 131,  213 

Distance  between  conductor  and  pole 158 

between  lightning  arresters 192 

between  overhead  conductors. 157 

Ducts,  standard  sizes  of  stoneware 207 

Flash-over  voltages  of  insulators 133,  134,  138,  140,  141 

Forces  on  wires,  resultant,  due  to  wind  and  ice 274 

Frequency,  natural,  of  line  (formula) 167 

Height  of  steel  towers  (formula) 39 

Horn  gaps,  settings  for 181 

405 


406  SUPPLEMENTARY  INDEX 

Inductance  between  power  and  telephone  lines  (formula) 124 

of  overhead  lines  (formulas) 82,  83,  165 

(effect  of  transposing  wires) 35 

(iron  wires) 87 

Insulators,  specification  for 393 

weight  of 336 

Iron  conductors  (formulas  and  curves) 85 

(example) 88 

Length  of  wire  in  span  (formulas) 254,  259 

as  influenced  by  temperature  and  tension  (chart) 264 

Line  drop  due  to  reactance  (iron  wires) 87 

economic 52 

(example) 56 

Loading  of  wires  due  to  wind  and  ice  (factor  ri) 275,  276,  277 

Losses  due  to  capacity  current  (example) 31 

corona 152,    154 

in  any  polyphase  system •.  .     22 

in  three-phase  line  (formula) Ill 

(example) 112 

in  underground  cables  (example) 224 

Mershon  diagram 90 

Modulus  of  elasticity 75,  279 

Opening  in  wall,  size  of,  for  entering  wires 144 

Pole  foundations,  depth  of 321.  378 

Poles,  concrete,  weight  of 325 

load  on  corner  (formula) 322 

wood,  deflection  of  (formula) 319 

physical  constants  for 314 

strength  of  (formula) 316 

volume  and  weight  of 314 

Power  factor  of  circuits  in  parallel 120 

of  underground  cables 224 

losses.     (See  Losses.) 
Pressure.     (See  Voltage.) 

Reactance.     (See  Line  drop;  Inductance.) 

Regulation,  voltage  (charts) 90,  93 

(formulas;  examples) 104 

Resistance  in  ground  connection  of  lightning  arrester  (example) 182 

insulation,  of  cables  (formula) 214 

of  conductors 76,  85 

Resonance,  frequency  of  (formula) 167 

Safety  factors  (insulators) 160 

Sag  (formulas) .,...,,.. 251,  258,  259 

(examples) 260 

Sag-temperature  calculations  (example) 286 

supports  at  different  elevations  (example) 293 


SUPPLEMENTARY  INDEX  407 

Skin  effect 77,  78 

with  iron  conductors  (curves) 86 

Sleet,  weight  of,  on  wires 265 

Spacing  of  conductors  (curves) 157 

of  lightning  arresters 192 

Sparking  distance  between  needle  points 160 

horn  gap  arresters 181 

Specification  for  steel  towers 391 

steel  tower  transmission  line 386 

wood  pole  transmission  line 375 

Specific  inductive  capacity  of  insulating  materials 131,  213 

resistance  of  cable  insulation 214 

Surge  impedance  (formula) 165 

Telephone  circuit;  inductance  formula 124 

Temperature-elongation  coefficient 279 

Temperature,  "critical"  (chart) 286 

rise  of  insulated  cables 226,  227 

Temperature-sag  calculations  (example) 286 

with  supports  at  different  elevations  (example) 293 

Tension  in  overhead  wires  (formulas) 252,  253,  256,  278 

Towers,  steel,  design  formulas 338 

example  of  design 340 

flexible  type,  example  of  deflection  calculations 352 

foundations,  weight  of '. . . . 345 

specification  for 391 

stiffness  (formula) 344 

Voltage  drop.     (See  Line  drop.) 

economic  (example) 63 

line  (formula) J 54 

rise  due  to  capacity  (formula) 109 

Volts,  inductive  (formulas),  82.     (See  also  Inductance.) 

Wind  pressure 268,  269,  274 

velocity  at  different  heights  from  ground  level 270 

Wire,  iron 85,  88 

length  of,  in  span 254,  259,  264 

table  (copper  and  aluminum) 76 

(physical  constants) 75 

Wires,  clearance  between,  and  pole 158 

for  entering 144 

distance  between  overhead 157 

economic  cross-section  of 53,  55 

loading  assumptions  for  mechanical  calculations 271 

due  to  wind  and  ice  (chart  for  factor  n) 275,  276 

sag  of 251,  258,  259 

sag-temperature  calculations 286,  293 

Wood  poles.     (See  Poles,  wood.) 

Young's  modulus 75,  279 


MESA 


A     000587292     4 


10571 
3:2- *-f 


